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References Bibliography

[1]
R. Ahlswede and D. E. Daykin, An inequality for the weights of two families of sets, their unions and intersections, Z. Wahrsch. Verw. Gebiete 43 (1978) no. 3, pp. 183-185.
[2]
M. Ajtai, J. Komlós and E. Szemerédi, A note on Ramsey numbers, J. Combin. Theory Ser. A 29 (1980) no. 3, pp. 354-360.
[3]
M. Ajtai and E. Szemerédi, Sets of lattice points that form no squares, Studia Sci. Math. Hungar. 9 (1974) pp. 9-11 (1975).
[4]
N. Alon, Independent sets in regular graphs and sum-free subsets of finite groups, Israel J. Math. 73 (1991) no. 2, pp. 247-256.
[5]
N. Alon, Probabilitistic methods in extremal finite set theory, 3 (1994) pp. 39-57. Extremal problems for finite sets (Visegrád, 1991) Bolyai Soc. Math. Stud. János Bolyai Math. Soc., Budapest;
[6]
N. Alon, The Shannon capacity of a union, Combinatorica 18 (1998) no. 3, pp. 301-310.
[7]
N. Alon, J. Balogh, B. Bollobás and R. Morris, The structure of almost all graphs in a hereditary property, J. Combin. Theory Ser. B 101 (2011) no. 2, pp. 85-110.
[8]
N. Alon, R. A. Duke, H. Lefmann, V. Rödl and R. Yuster, The algorithmic aspects of the regularity lemma, J. Algorithms 16 (1994) no. 1, pp. 80-109.
[9]
N. Alon, D. Fischer, M. Krivelevich and M. Szegedy, Efficient testing of large graphs, Combinatorica 20 (2000) no. 4, pp. 451-476.
[10]
N. Alon and S. Friedland, The maximum number of perfect matchings in graphs with a given degree sequence, Electron. J. Combin. 15 (2008) no. 1, Note 13, 2.
[11]
N. Alon, A. Moitra and B. Sudakov, Nearly complete graphs decomposable into large induced matchings and their applications, J. Eur. Math. Soc. (JEMS) 15 (2013) no. 5, pp. 1575-1596.
[12]
N. Alon, L. Rónyai and T. Szabó, Norm-graphs: variations and applications, J. Combin. Theory Ser. B 76 (1999) no. 2, pp. 280-290.
[13]
N. Alon and J. H. Spencer, The probabilistic method, (2016) Wiley Series in Discrete Mathematics and Optimization John Wiley & Sons, Inc., Hoboken, NJ; Fourth
[14]
R. Alweiss, B. Huang and M. Sellke, Improved Lower Bound for the Union-Closed Sets Conjecture, (2022) arXiv:2211.11731v1
[15]
R. Alweiss, S. Lovett, K. Wu and J. Zhang, Improved bounds for the sunflower lemma, Ann. of Math. (2) 194 (2021) no. 3, pp. 795-815.
[16]
I. Anderson, Combinatorics of finite sets, (2002) Corrected reprint of the 1989 edition Dover Publications, Inc., Mineola, NY;
[17]
I. Anderson, On the divisors of a number, J. London Math. Soc. 43 (1968) pp. 410-418.
[18]
R. P. Anstee, L. Rónyai and A. Sali, Shattering news, Graphs Combin. 18 (2002) no. 1, pp. 59-73.
[19]
L. Babai, Honors Combinatorics CMSC-27410/37200 Midterm, (2012)
[20]
L. Babai, Honors Combinatorics CMSC-27410/Math-28400/CMSC-37200 Second Midterm, (2018)
[21]
J. Balogh, Math 595 - Container Method, (Accessed on July 30, 2020)
[22]
J. Balogh, H. Liu and M. Sharifzadeh, The number of subsets of integers with no \(k\)-term arithmetic progression, Int. Math. Res. Not. IMRN (2017) no. 20, pp. 6168-6186.
[23]
J. Balogh, R. Morris and W. Samotij, Independent sets in hypergraphs, J. Amer. Math. Soc. 28 (2015) no. 3, pp. 669-709.
[24]
J. Balogh, R. Mycroft and A. Treglown, A random version of Sperner’s theorem, J. Combin. Theory Ser. A 128 (2014) pp. 104-110.
[25]
J. Balogh, A. Treglown and A. Zs. Wagner, Applications of graph containers in the Boolean lattice, Random Structures Algorithms 49 (2016) no. 4, pp. 845-872.
[26]
J. Balogh and A. Zs. Wagner, Kleitman’s conjecture about families of given size minimizing the number of \(k\)-chains, Adv. Math. 330 (2018) pp. 229-252.
[27]
F. A. Behrend, On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. U.S.A. 32 (1946) pp. 331-332.
[28]
P. V. M. Blagojević, B. Bukh and R. Karasev, Turán numbers for \(K_s,t\)-free graphs: topological obstructions and algebraic constructions, Israel J. Math. 197 (2013) no. 1, pp. 199-214.
[29]
G. R. Blakley and P. Roy, A Hölder type inequality for symmetric matrices with nonnegative entries, Proc. Amer. Math. Soc. 16 (1965) pp. 1244-1245.
[30]
G. Blekherman and A. Raymond, Proof of the Erdős-Simonovits Conjecture on Walks, (2020) arXiv:2009.10845v1
[31]
G. Blekherman, A. Raymond, M. Singh and R. R. Thomas, Simple graph density inequalities with no sum of squares proofs, Combinatorica 40 (2020) no. 4, pp. 455-471.
[32]
T. F. Bloom and O. Sisask, Breaking the logarithmic barrier in Roth’s theorem on arithmetic progressions, (2020) arXiv:2007.03528v1
[33]
T. Bohman, A limit theorem for the Shannon capacities of odd cycles. I, Proc. Amer. Math. Soc. 131 (2003) no. 11, pp. 3559-3569.
[34]
T. Bohman, The triangle-free process, Adv. Math. 221 (2009) no. 5, pp. 1653-1677.
[35]
T. Bohman and P. Keevash, Dynamic concentration of the triangle-free process, Random Structures Algorithms 58 (2021) no. 2, pp. 221-293.
[36]
B. Bollobás, Combinatorics, (1986) Set systems, hypergraphs, families of vectors and combinatorial probability Cambridge University Press, Cambridge;
[37]
B. Bollobás, Extremal graph theory, (2004) Reprint of the 1978 original Dover Publications, Inc., Mineola, NY;
[38]
B. Bollobás, Modern graph theory, 184 (1998) Graduate Texts in Mathematics Springer-Verlag, New York;
[39]
B. Bollobás, On generalized graphs, Acta Math. Acad. Sci. Hungar. 16 (1965) pp. 447-452.
[40]
B. Bollobás, Relations between sets of complete subgraphs, (1976) pp. 79-84. Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975) Congressus Numerantium, No. XV Utilitas Math., Winnipeg, Man.;
[41]
B. Bollobás, The work of William Timothy Gowers, Doc. Math. (1998) no. Extra Vol. I, pp. 109-118. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998)
[42]
J. A. Bondy and M. Simonovits, Cycles of even length in graphs, J. Combinatorial Theory Ser. B 16 (1974) pp. 97-105.
[43]
L. M. Brègman, Certain properties of nonnegative matrices and their permanents, Dokl. Akad. Nauk SSSR 211 (1973) pp. 27-30.
[44]
W. G. Brown, On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9 (1966) pp. 281-285.
[45]
M. Bucić, S. Letzter, B. Sudakov and T. Tran, Minimum saturated families of sets, Bull. Lond. Math. Soc. 50 (2018) no. 4, pp. 725-732.
[46]
B. Bukh, Random algebraic construction of extremal graphs, Bull. Lond. Math. Soc. 47 (2015) no. 6, pp. 939-945.
[47]
B. Bukh and D. Conlon, Rational exponents in extremal graph theory, J. Eur. Math. Soc. (JEMS) 20 (2018) no. 7, pp. 1747-1757.
[48]
M. Campos, L. Mattos, R. Morris and N. Morrison, On the singularity of random symmetric matrices, Duke Math. J. 170 (2021) no. 5, pp. 881-907.
[49]
Z. Chase and S. Lovett, Approximate union closed conjecture, (2022) arXiv:2211.11689v1
[50]
M. Chudnovsky, R. Kim, C.-H. Liu, P. Seymour and S. Thomassé, Domination in tournaments, J. Combin. Theory Ser. B 130 (2018) pp. 98-113.
[51]
F. R. K. Chung, R. L. Graham, P. Frankl and J. B. Shearer, Some intersection theorems for ordered sets and graphs, J. Combin. Theory Ser. A 43 (1986) no. 1, pp. 23-37.
[52]
F. R. K. Chung, R. L. Graham and R. M. Wilson, Quasi-random graphs, Combinatorica 9 (1989) no. 4, pp. 345-362.
[53]
D. Conlon, A new upper bound for diagonal Ramsey numbers, Ann. of Math. (2) 170 (2009) no. 2, pp. 941-960.
[54]
D. Conlon, Extremal graph theory, (Accessed on April 21, 2020)
[55]
D. Conlon, Graphs with few paths of prescribed length between any two vertices, Bull. Lond. Math. Soc. 51 (2019) no. 6, pp. 1015-1021.
[56]
D. Conlon and J. Fox, Bounds for graph regularity and removal lemmas, Geom. Funct. Anal. 22 (2012) no. 5, pp. 1191-1256.
[57]
D. Conlon and J. Fox, Graph removal lemmas, 409 (2013) pp. 1-49. Surveys in combinatorics 2013 London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge;
[58]
D. Conlon, J. Fox and B. Sudakov, An approximate version of Sidorenko’s conjecture, Geom. Funct. Anal. 20 (2010) no. 6, pp. 1354-1366.
[59]
D. Conlon, J. Fox, B. Sudakov and Y. Zhao, The regularity method for graphs with few 4-cycles, J. Lond. Math. Soc. (2) 104 (2021) no. 5, pp. 2376-2401.
[60]
D. Conlon, J. H. Kim, C. Lee and J. Lee, Some advances on Sidorenko’s conjecture, J. Lond. Math. Soc. (2) 98 (2018) no. 3, pp. 593-608.
[61]
D. Conlon and J. Lee, Sidorenko’s conjecture for blow-ups, Discrete Anal. (2021) Paper No. 2, 13.
[62]
J. W. Cooper, D. Kráľ and T. L. Martins, Finitely forcible graph limits are universal, Adv. Math. 340 (2018) pp. 819-854.
[63]
G. Coutinho, Semidefinite programming (and graph theory), (Accessed on August 17, 2023)
[64]
J. Cutler and A. J. Radcliffe, An entropy proof of the Kahn-Lovász theorem, Electron. J. Combin. 18 (2011) no. 1, Paper 10, 9.
[65]
J. Cutler and A. J. Radcliffe, Minimizing the number of independent sets in triangle-free regular graphs, Discrete Math. 341 (2018) no. 3, pp. 793-800.
[66]
S. Das, Applications of the Szemerédi Regularity Lemma, FU Berlin, (Accessed on November 2, 2020)
[67]
S. Das, W. Gan and B. Sudakov, Sperner’s theorem and a problem of Erdős, Katona and Kleitman, Combin. Probab. Comput. 24 (2015) no. 4, pp. 585-608.
[68]
E. Davies, M. Jenssen and W. Perkins, A proof of the upper matching conjecture for large graphs, J. Combin. Theory Ser. B 151 (2021) pp. 393-416.
[69]
E. Davies, M. Jenssen, W. Perkins and B. Roberts, Independent sets, matchings, and occupancy fractions, J. Lond. Math. Soc. (2) 96 (2017) no. 1, pp. 47-66.
[70]
E. Davies, M. Jenssen, W. Perkins and B. Roberts, On the average size of independent sets in triangle-free graphs, Proc. Amer. Math. Soc. 146 (2018) no. 1, pp. 111-124.
[71]
N. G. de Bruijn, Ca. van Ebbenhorst Tengbergen and D. Kruyswijk, On the set of divisors of a number, Nieuw Arch. Wiskunde (2) 23 (1951) pp. 191-193.
[72]
D. de Caen, A note on the probabilistic approach to Turán’s problem, J. Combin. Theory Ser. B 34 (1983) no. 3, pp. 340-349.
[73]
R. Diestel, Graph theory, 173 (2005) Graduate Texts in Mathematics Springer-Verlag, Berlin; Third
[74]
I. Dinur and S. Safra, The importance of being biased, (2002) pp. 33-42. Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing ACM, New York;
[75]
A. P. Dove, J. R. Griggs, R. J. Kang and J.-S. Sereni, Supersaturation in the Boolean lattice, Integers 14A (2014) Paper No. A4, 7.
[76]
D. P. Dubhashi and A. Panconesi, Concentration of measure for the analysis of randomized algorithms, (2009) Cambridge University Press, Cambridge;
[77]
P. Dukes, Notes for Math 422: Enumeration and Ramsey Theory, University of Victoria, (Accessed on April 7, 2020)
[78]
S. Durocher, D. S. Gunderson, P. C. Li and M. Skala, Cycle-maximal triangle-free graphs, Discrete Math. 338 (2015) no. 2, pp. 274-290.
[79]
Z. Dvořák, N. Morrison, J. A. Noel, S. Norin and L. Postle, Bounding the number of cycles in a graph in terms of its degree sequence, European J. Combin. 91 (2021) Paper No. 103206, 16.
[80]
G. P. Egoryčev, \cyr Reshenie problemy van-\cyr der-\cyr Vardena dlya permanentov, (1980) 12. Preprint IFSO-13 M Akad. Nauk SSSR Sibirsk. Otdel., Inst. Fiz., Krasnoyarsk;
[81]
D. Ellis, Y. Filmus and E. Friedgut, Triangle-intersecting families of graphs, J. Eur. Math. Soc. (JEMS) 14 (2012) no. 3, pp. 841-885.
[82]
D. Ellis, E. Friedgut and H. Pilpel, Intersecting families of permutations, J. Amer. Math. Soc. 24 (2011) no. 3, pp. 649-682.
[83]
K. Engel, Sperner theory, 65 (1997) Encyclopedia of Mathematics and its Applications Cambridge University Press, Cambridge;
[84]
P. Erdős, On a combinatorial problem, Nordisk Mat. Tidskr. 11 (1963) pp. 5-10, 40.
[85]
P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. 51 (1945) pp. 898-902.
[86]
P. Erdős, On a theorem of Rademacher-Turán, Illinois J. Math. 6 (1962) pp. 122-127.
[87]
P. Erdős, On some new inequalities concerning extremal properties of graphs, (1968) pp. 77-81. Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic Press, New York;
[88]
P. Erdős, On the graph theorem of Turán, Mat. Lapok 21 (1970) pp. 249-251 (1971).
[89]
P. Erdős, Problems and results in combinatorial analysis and combinatorial number theory, (1991) pp. 397-406. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) Wiley-Intersci. Publ. Wiley, New York;
[90]
P. Erdős, Problems and results in combinatorial analysis and graph theory, Discrete Math. 72 (1988) no. 1-3, pp. 81-92. Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986)
[91]
P. Erdős, Some recent results on extremal problems in graph theory. Results, (1967) pp. 117-123 (English); pp. 124--130 (French). Theory of Graphs (Internat. Sympos., Rome, 1966) Gordon and Breach, New York; Dunod, Paris;
[92]
P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947) pp. 292-294.
[93]
P. Erdős, E. Győri and M. Simonovits, How many edges should be deleted to make a triangle-free graph bipartite?, 60 (1992) pp. 239-263. Sets, graphs and numbers (Budapest, 1991) Colloq. Math. Soc. János Bolyai North-Holland, Amsterdam;
[94]
P. Erdős and A. Hajnal, Ramsey-type theorems, Discrete Appl. Math. 25 (1989) no. 1-2, pp. 37-52. Combinatorics and complexity (Chicago, IL, 1987)
[95]
P. Erdős, A. Hajnal and J. W. Moon, A problem in graph theory, Amer. Math. Monthly 71 (1964) pp. 1107-1110.
[96]
P. Erdős and D. J. Kleitman, Extremal problems among subsets of a set, Discrete Math. 8 (1974) pp. 281-294.
[97]
P. Erdős, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2) 12 (1961) pp. 313-320.
[98]
P. Erdős and R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 (1960) pp. 85-90.
[99]
P. Erdős and M. Simonovits, A limit theorem in graph theory, Studia Sci. Math. Hungar. 1 (1966)
[100]
P. Erdős and M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982) no. 3, pp. 275-288.
[101]
P. Erdős and M. Simonovits, Cube-supersaturated graphs and related problems, (1984) pp. 203-218. Progress in graph theory (Waterloo, Ont., 1982) Academic Press, Toronto, ON;
[102]
P. Erdős and M. Simonovits, Supersaturated graphs and hypergraphs, Combinatorica 3 (1983) no. 2, pp. 181-192.
[103]
P. Erdős and A. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946) pp. 1087-1091.
[104]
P. Erdős and P. Turán, On Some Sequences of Integers, J. London Math. Soc. 11 (1936) no. 4, pp. 261-264.
[105]
Z. Füredi, A proof of the stability of extremal graphs, Simonovits’ stability from Szemerédi’s regularity, J. Combin. Theory Ser. B 115 (2015) pp. 66-71.
[106]
Z. Füredi, Extremal hypergraphs and combinatorial geometry, (1995) pp. 1343-1352. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel;
[107]
Z. Füredi, New asymptotics for bipartite Turán numbers, J. Combin. Theory Ser. A 75 (1996) no. 1, pp. 141-144.
[108]
Z. Füredi, A. Naor and J. Verstraëte, On the Turán number for the hexagon, Adv. Math. 203 (2006) no. 2, pp. 476-496.
[109]
Z. Füredi and M. Simonovits, The history of degenerate (bipartite) extremal graph problems, 25 (2013) pp. 169-264. Erdös centennial Bolyai Soc. Math. Stud. János Bolyai Math. Soc., Budapest;
[110]
D. I. Falikman, Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix, Mat. Zametki 29 (1981) no. 6, pp. 931-938, 957.
[111]
Y. Filmus, Spectral Methods for Intersection Problems: Friedgut’s Research Program in Extremal Combinatorics, (2012, Accessed on April 10, 2020)
[112]
D. C. Fisher, Lower bounds on the number of triangles in a graph, J. Graph Theory 13 (1989) no. 4, pp. 505-512.
[113]
G. Fiz Pontiveros, S. Griffiths and R. Morris, The Triangle-Free Process and the Ramsey Number \(R(3,k)\), Mem. Amer. Math. Soc. 263 (2020) no. 1274, 0.
[114]
M. M. Fleck, Building Blocks for Theoretical Computer Science. Version 1.3, (Accessed on April 15, 2020)
[115]
C. M. Fortuin, P. W. Kasteleyn and J. Ginibre, Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971) pp. 89-103.
[116]
J. Fox, A new proof of the graph removal lemma, Ann. of Math. (2) 174 (2011) no. 1, pp. 561-579.
[117]
J. Fox, MAT 307: Combinatorics, (Accessed on April 14, 2020)
[118]
J. Fox, H. Huang and B. Sudakov, On graphs decomposable into induced matchings of linear sizes, Bull. Lond. Math. Soc. 49 (2017) no. 1, pp. 45-57.
[119]
J. Fox and L. M. Lovász, A tight lower bound for Szemerédi’s regularity lemma, Combinatorica 37 (2017) no. 5, pp. 911-951.
[120]
J. Fox and B. Sudakov, Dependent random choice, Random Structures Algorithms 38 (2011) no. 1-2, pp. 68-99.
[121]
P. Frankl, Extremal set systems, (1995) pp. 1293-1329. Handbook of combinatorics, Vol. 1, 2 Elsevier Sci. B. V., Amsterdam;
[122]
P. Frankl and R. L. Graham, Old and new proofs of the Erdős-Ko-Rado theorem, Sichuan Daxue Xuebao 26 (1989) no. Special Issue, pp. 112-122.
[123]
P. Frankl and N. Tokushige, Invitation to intersection problems for finite sets, J. Combin. Theory Ser. A 144 (2016) pp. 157-211.
[124]
E. Friedgut, Proof of an intersection theorem via Fourier analysis, (2005) pp. 117-120. European Congress of Mathematics Eur. Math. Soc., Zürich;
[125]
S. Friedland, E. Krop and K. Markström, On the number of matchings in regular graphs, Electron. J. Combin. 15 (2008) no. 1, Research Paper 110, 28.
[126]
A. Frieze and R. Kannan, Quick approximation to matrices and applications, Combinatorica 19 (1999) no. 2, pp. 175-220.
[127]
H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978) pp. 275-291 (1979).
[128]
D. Galvin, Counting colorings of a regular graph, Graphs Combin. 31 (2015) no. 3, pp. 629-638.
[129]
D. Galvin, Independent sets in the discrete hypercube, (2019) arXiv:1901.01991v1
[130]
D. Galvin, On homomorphisms from the Hamming cube to \(\mathbbZ\), Israel J. Math. 138 (2003) pp. 189-213.
[131]
D. Galvin, Three tutorial lectures on entropy and counting, (2014) arXiv:1406.7872v1
[132]
D. Gerbner, B. Keszegh, N. Lemons, C. Palmer, D. Pálvölgyi and B. Patkós, Saturating Sperner families, Graphs Combin. 29 (2013) no. 5, pp. 1355-1364.
[133]
D. Gerbner and B. Patkós, Extremal finite set theory, (2019) Discrete Mathematics and its Applications (Boca Raton) CRC Press, Boca Raton, FL;
[134]
J. Gilmer, A constant lower bound for the union-closed sets conjecture, (2022) arXiv:2211.09055v1
[135]
O. Goldreich, S. Goldwasser and D. Ron, Property testing and its connection to learning and approximation, J. ACM 45 (1998) no. 4, pp. 653-750.
[136]
A. W. Goodman, On sets of acquaintances and strangers at any party, Amer. Math. Monthly 66 (1959) pp. 778-783.
[137]
W. T. Gowers, Entropy and Sidorenko’s conjecture -- after Szegedy, , (2015, Accessed on April 28, 2020)
[138]
W. T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. of Math. (2) 166 (2007) no. 3, pp. 897-946.
[139]
W. T. Gowers, Lower bounds of tower type for Szemerédi’s uniformity lemma, Geom. Funct. Anal. 7 (1997) no. 2, pp. 322-337.
[140]
J. E. Greene, A new short proof of Kneser’s conjecture, Amer. Math. Monthly 109 (2002) no. 10, pp. 918-920.
[141]
D. Gunderson, Graph Theory Notes, University of Manitoba, (Accessed on December 21, 2022)
[142]
G. Halász, Estimates for the concentration function of combinatorial number theory and probability, Period. Math. Hungar. 8 (1977) no. 3-4, pp. 197-211.
[143]
T. S. Han, Nonnegative entropy measures of multivariate symmetric correlations, Information and Control 36 (1978) no. 2, pp. 133-156.
[144]
G. Hansel, Sur le nombre des fonctions booléennes monotones de \(n\) variables, C. R. Acad. Sci. Paris Sér. A-B 262 (1966) pp. A1088-A1090.
[145]
T. E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc. 56 (1960) pp. 13-20.
[146]
H. Hatami, Graph norms and Sidorenko’s conjecture, Israel J. Math. 175 (2010) pp. 125-150.
[147]
H. Hatami and S. Norine, Undecidability of linear inequalities in graph homomorphism densities, J. Amer. Math. Soc. 24 (2011) no. 2, pp. 547-565.
[148]
P. Hell and J. Nešetřil, Graphs and homomorphisms, 28 (2004) Oxford Lecture Series in Mathematics and its Applications Oxford University Press, Oxford;
[149]
T. Jiang, J. Ma and L. Yepremyan, On Turán exponents of bipartite graphs, Combin. Probab. Comput. 31 (2022) no. 2, pp. 333-344.
[150]
S. Jukna, Extremal combinatorics, (2011) With applications in computer science Texts in Theoretical Computer Science. An EATCS Series Springer, Heidelberg; Second
[151]
T. Kővari, V. T. Sós and P. Turán, On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954) pp. 50-57.
[152]
J. Kahn, Entropy, independent sets and antichains: a new approach to Dedekind’s problem, Proc. Amer. Math. Soc. 130 (2002) no. 2, pp. 371-378.
[153]
G. Kalai, Extremal Combinatorics I: Extremal Problems on Set Systems, , (2008, Accessed on April 10, 2020)
[154]
G. Kalai, Extremal Combinatorics III: Some Basic Theorems, , (2008, Accessed on April 10, 2020)
[155]
G. Kalai, Polymath10: The Erdős Rado Delta System Conjecture, , (2015, Accessed on April 10, 2020)
[156]
G. Kalai, To cheer you up in difficult times 7: Bloom and Sisask just broke the logarithm barrier for Roth’s theorem!, , (2020, Accessed on July 30, 2020)
[157]
D. Y. Kang, J. Kim and H. Liu, On the rational Turán exponents conjecture, J. Combin. Theory Ser. B 148 (2021) pp. 149-172.
[158]
R. J. Kang and T. Müller, Probabilistic and extremal combinatorics, (Accessed on April 14, 2020)
[159]
G. Katona, A theorem of finite sets, (1968) pp. 187-207. Theory of graphs (Proc. Colloq., Tihany, 1966)
[160]
G. Katona, On a conjecture of Erdős and a stronger form of Sperner’s theorem, Studia Sci. Math. Hungar. 1 (1966) pp. 59-63.
[161]
G. O. H. Katona, A simple proof of the Erdős-Chao Ko-Rado theorem, J. Combinatorial Theory Ser. B 13 (1972) pp. 183-184.
[162]
P. Keevash, Hypergraph Turán problems, 392 (2011) pp. 83-139. Surveys in combinatorics 2011 London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge;
[163]
P. Keevash and B. Sudakov, The Turán number of the Fano plane, Combinatorica 25 (2005) no. 5, pp. 561-574.
[164]
J. H. Kim, C. Lee and J. Lee, Two approaches to Sidorenko’s conjecture, Trans. Amer. Math. Soc. 368 (2016) no. 7, pp. 5057-5074.
[165]
Jeong Han Kim, The Ramsey number \(R(3,t)\) has order of magnitude \(t^2/\log t\), Random Structures Algorithms 7 (1995) no. 3, pp. 173-207.
[166]
D. Kleitman, A conjecture of Erdős-Katona on commensurable pairs among subsets of an \(n\)-set, (1968) pp. 215-218. Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic Press, New York;
[167]
D. Kleitman, On Dedekind’s problem: The number of monotone Boolean functions, Proc. Amer. Math. Soc. 21 (1969) pp. 677-682.
[168]
D. J. Kleitman, Families of non-disjoint subsets, J. Combinatorial Theory 1 (1966) pp. 153-155.
[169]
D. J. Kleitman, On a lemma of Littlewood and Offord on the distribution of certain sums, Math. Z. 90 (1965) pp. 251-259.
[170]
D. J. Kleitman, On a lemma of Littlewood and Offord on the distributions of linear combinations of vectors, Advances in Math. 5 (1970) pp. 155-157 (1970).
[171]
D. J. Kleitman, On an extremal property of antichains in partial orders. The LYM property and some of its implications and applications, (1974) pp. 77-90. Math. Centre Tracts, No. 56. Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974), Part 2: Graph theory; foundations, partitions and combinatorial geometry
[172]
D. J. Kleitman and K. J. Winston, On the number of graphs without \(4\)-cycles, Discrete Math. 41 (1982) no. 2, pp. 167-172.
[173]
D. J. Kleitman and K. J. Winston, The asymptotic number of lattices, Ann. Discrete Math. 6 (1980) pp. 243-249.
[174]
M. Kneser, Aufgabe 360, Jahresbericht der Deutschen Mathematiker-Vereinigung 58 (1956) no. 2, 27.
[175]
J. Kollár, L. Rónyai and T. Szabó, Norm-graphs and bipartite Turán numbers, Combinatorica 16 (1996)
[176]
J. Komlós, Tiling Turán theorems, Combinatorica 20 (2000) no. 2, pp. 203-218.
[177]
S. Kopparty and B. Rossman, The homomorphism domination exponent, European J. Combin. 32 (2011) no. 7, pp. 1097-1114.
[178]
D. Korándi, Graph Theory 2018, (Accessed on April 6, 2020)
[179]
D. Korándi, A. Roberts and A. Scott, Exact stability for Turán’s theorem, Adv. Comb. (2021) Paper No. 9, 17.
[180]
A. D. Korshunov, The number of monotone Boolean functions, Problemy Kibernet. (1981) no. 38, pp. 5-108, 272.
[181]
A. D. Korshunov and A. A. Sapozhenko, The number of binary codes with distance \(2\), Problemy Kibernet. (1983) no. 40, pp. 111-130.
[182]
A. Kostochka and L. Pyber, Small topological complete subgraphs of ``dense’’ graphs, Combinatorica 8 (1988) no. 1, pp. 83-86.
[183]
J. B. Kruskal, The number of simplices in a complex, (1963) pp. 251-278. Mathematical optimization techniques Univ. of California Press, Berkeley, Calif.;
[184]
M. Kwan and L. Sauermann, An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs, Discrete Anal. (2020) Paper No. 12, 34.
[185]
D. V. Le and T. Römer, A Kruskal-Katona type result and applications, Discrete Math. 343 (2020) no. 5, 111801, 12.
[186]
I. Leader, Combinatorics, 2010, University of Cambridge. Notes scribed by Gareth Taylor, (Accessed on April 6, 2020)
[187]
J. L. X. Li and B. Szegedy, On the logarithimic calculus and Sidorenko’s conjecture, (2011) arXiv:1107.1153v1
[188]
S. Y. R. Li, An extremal problem among subsets of a set, J. Combinatorial Theory Ser. A 23 (1977) no. 3, pp. 341-343.
[189]
B. Lidick\’y, Math 608 - Extremal Graph Theory, Iowa State University, (Accessed on July 30, 2020)
[190]
N. Linial, Minicourse on Geometry and Combinatorics at the Graphs and Groups Workshop, (Accessed on July 30, 2020)
[191]
N. Linial and Z. Luria, An upper bound on the number of high-dimensional permutations, Combinatorica 34 (2014) no. 4, pp. 471-486.
[192]
N. Linial and Y. Rabinovich, Local and global clique numbers, J. Combin. Theory Ser. B 61 (1994) no. 1, pp. 5-15.
[193]
J. H. van Lint and R. M. Wilson, A course in combinatorics, (2001) Cambridge University Press, Cambridge; Second
[194]
J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. III, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943) pp. 277-286.
[195]
H. Liu, Topics in extremal combinatorics, 2019, Shandong University, (Accessed on April 9, 2020)
[196]
H. Liu, O. Pikhurko and K. Staden, The exact minimum number of triangles in graphs with given order and size, Forum Math. Pi 8 (2020) e8, 144.
[197]
E. Long, Extremal Combinatorics, 2012, Queen Mary, University of London, (Accessed on April 9, 2020)
[198]
L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc 55 (1949) pp. 961-962.
[199]
L. Lovász, Combinatorial problems and exercises, (2007) AMS Chelsea Publishing, Providence, RI; second
[200]
L. Lovász, Kneser’s conjecture, chromatic number, and homotopy, J. Combin. Theory Ser. A 25 (1978) no. 3, pp. 319-324.
[201]
L. Lovász, Large networks and graph limits, 60 (2012) American Mathematical Society Colloquium Publications American Mathematical Society, Providence, RI;
[202]
L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979) no. 1, pp. 1-7.
[203]
L. Lovász and V. T. Sós, Generalized quasirandom graphs, J. Combin. Theory Ser. B 98 (2008) no. 1, pp. 146-163.
[204]
L. Lovász and M. Simonovits, On the number of complete subgraphs of a graph. II, (1983) pp. 459-495. Studies in pure mathematics Birkhäuser, Basel;
[205]
L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B 96 (2006) no. 6, pp. 933-957.
[206]
L. Lovász and B. Szegedy, Szemerédi’s lemma for the analyst, Geom. Funct. Anal. 17 (2007) no. 1, pp. 252-270.
[207]
L. Lovász and B. Szegedy, Testing properties of graphs and functions, Israel J. Math. 178 (2010) pp. 113-156.
[208]
D. Lubell, A short proof of Sperner’s lemma, J. Combinatorial Theory 1 (1966) 299.
[209]
L. Lusternik and L. Schnirelmann, Méthodes topologiques dans les problèmes variationnels, Moscow: Gosudarstvennoe Izdat
[210]
N. Lyall, Behrend’s Example, (Accessed on August 16, 2023)
[211]
R. Lyons, Asymptotic enumeration of spanning trees, Combin. Probab. Comput. 14 (2005) no. 4, pp. 491-522.
[212]
G. MacGillivray and P. Dukes, Notes for Math 322: Intermediate Combinatorics, University of Victoria, (Accessed on April 7, 2020)
[213]
W. Mantel, Problem 28, Wiskundige Opgaven 10 (1907) pp. 60-61.
[214]
J. Matoušek, Using the Borsuk-Ulam theorem, (2003) Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler Universitext Springer-Verlag, Berlin;
[215]
S. Mattheus and J. Verstraëte, The asymptotics of \(r(4,t)\), (2023) arXiv:2306.04007v4
[216]
L. D. Mešalkin, A generalization of Sperner’s theorem on the number of subsets of a finite set, Teor. Verojatnost. i Primenen 8 (1963) pp. 219-220.
[217]
S. Messuti and M. Schacht, Extremale Graphentheorie, (Accessed on April 23, 2020)
[218]
H. Minc, Upper bounds for permanents of \((0,\,1)\)-matrices, Bull. Amer. Math. Soc. 69 (1963) pp. 789-791.
[219]
L. Mirsky, A dual of Dilworth’s decomposition theorem, Amer. Math. Monthly 78 (1971) pp. 876-877.
[220]
M. Molloy and B. Reed, Graph colouring and the probabilistic method, 23 (2002) Algorithms and Combinatorics Springer-Verlag, Berlin;
[221]
N. Morrison, J. A. Noel and A. Scott, On saturated \(k\)-Sperner systems, Electron. J. Combin. 21 (2014) no. 3, Paper 3.22, 17.
[222]
G. Moshkovitz and A. Shapira, A short proof of Gowers’ lower bound for the regularity lemma, Combinatorica 36 (2016) no. 2, pp. 187-194.
[223]
T. S. Motzkin and E. G. Straus, Maxima for graphs and a new proof of a theorem of Turán, Canadian J. Math. 17 (1965) pp. 533-540.
[224]
H. H. Nguyen and V. H. Vu, Small ball probability, inverse theorems, and applications, 25 (2013) pp. 409-463. Erdös centennial Bolyai Soc. Math. Stud. János Bolyai Math. Soc., Budapest;
[225]
V. Nikiforov, The number of cliques in graphs of given order and size, Trans. Amer. Math. Soc. 363 (2011) no. 3, pp. 1599-1618.
[226]
J. A. Noel, A. Scott and B. Sudakov, Supersaturation in posets and applications involving the container method, J. Combin. Theory Ser. A 154 (2018) pp. 247-284.
[227]
S. Norin, Math 550 - Combinatorics, 2018, McGill University. Notes scribed by Eric P. Hanson, (Accessed on April 6, 2020)
[228]
S. Norin and L. Yepremyan, Turán number of generalized triangles, J. Combin. Theory Ser. A 146 (2017) pp. 312-343.
[229]
D. Osthus, Maximum antichains in random subsets of a finite set, J. Combin. Theory Ser. A 90 (2000) no. 2, pp. 336-346.
[230]
G. Pólya and G. Szegő, Problems and theorems in analysis. I, (1998) Series, integral calculus, theory of functions, Translated from the German by Dorothee Aeppli, Reprint of the 1978 English translation Classics in Mathematics Springer-Verlag, Berlin;
[231]
L. Pebody, Extension of a Method of Gilmer, (2022) arXiv:2211.13139v1
[232]
O. Pikhurko, A note on the Turán function of even cycles, Proc. Amer. Math. Soc. 140 (2012) no. 11, pp. 3687-3692.
[233]
O. Pikhurko, An exact Turán result for the generalized triangle, Combinatorica 28 (2008) no. 2, pp. 187-208.
[234]
D. H. J. Polymath, A new proof of the density Hales-Jewett theorem, Ann. of Math. (2) 175 (2012) no. 3, pp. 1283-1327.
[235]
V. Rödl and J. Skokan, Regularity lemma for \(k\)-uniform hypergraphs, Random Structures Algorithms 25 (2004) no. 1, pp. 1-42.
[236]
J. Radhakrishnan, An entropy proof of Bregman’s theorem, J. Combin. Theory Ser. A 77 (1997) no. 1, pp. 161-164.
[237]
F. P. Ramsey, On a Problem of Formal Logic, Proc. London Math. Soc. (2) 30 (1929) no. 4, pp. 264-286.
[238]
A. Rao, Coding for sunflowers, Discrete Anal. (2020) Paper No. 2, 8.
[239]
A. A. Razborov, Flag algebras, J. Symbolic Logic 72 (2007) no. 4, pp. 1239-1282.
[240]
A. A. Razborov, On the minimal density of triangles in graphs, Combin. Probab. Comput. 17 (2008) no. 4, pp. 603-618.
[241]
C. Reiher, The clique density theorem, Ann. of Math. (2) 184 (2016) no. 3, pp. 683-707.
[242]
J. Rollins, Extremal Combinatorics Problem Database. Karlsruhe Institute of Technology.,
[243]
K. F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953) pp. 104-109.
[244]
M. Rudelson and R. Vershynin, The Littlewood-Offord problem and invertibility of random matrices, Adv. Math. 218 (2008) no. 2, pp. 600-633.
[245]
I. Z. Ruzsa and E. Szemerédi, Triple systems with no six points carrying three triangles, 18 (1978) pp. 939-945. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II Colloq. Math. Soc. János Bolyai North-Holland, Amsterdam-New York;
[246]
V. T. Sós, P. Erdős and W. G. Brown, On the existence of triangulated spheres in \(3\)-graphs, and related problems, Period. Math. Hungar. 3 (1973) no. 3-4, pp. 221-228.
[247]
M. Sağlam, Near log-convexity of measured heat in (discrete) time and consequences, (2018) pp. 967-978. 59th Annual IEEE Symposium on Foundations of Computer Science---FOCS 2018 IEEE Computer Soc., Los Alamitos, CA;
[248]
W. Samotij, Counting independent sets in graphs, European J. Combin. 48 (2015) pp. 5-18.
[249]
W. Samotij, Subsets of posets minimising the number of chains, Trans. Amer. Math. Soc. 371 (2019) no. 10, pp. 7259-7274.
[250]
A. A. Sapozhenko, On the number of independent sets in extenders, Diskret. Mat. 13 (2001) no. 1, pp. 56-62.
[251]
A. A. Sapozhenko, The Cameron-Erdős conjecture, Dokl. Akad. Nauk 393 (2003) no. 6, pp. 749-752.
[252]
N. Sauer, On the density of families of sets, J. Combinatorial Theory Ser. A 13 (1972) pp. 145-147.
[253]
W. Sawin, An improved lower bound for the union-closed set conjecture, (2022) arXiv:2211.11504v1
[254]
D. Saxton and A. Thomason, Hypergraph containers, Invent. Math. 201 (2015) no. 3, pp. 925-992.
[255]
A. Schrijver, A short proof of Minc’s conjecture, J. Combinatorial Theory Ser. A 25 (1978) no. 1, pp. 80-83.
[256]
I. Schur, Über Kongruenz \(x^m+y^m=z^m(\bmod p)\), Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1916) pp. 114-116.
[257]
A. Scott, C8.3 Combinatorics, 2020, University of Oxford. Modified by J. Long, (Accessed on April 6, 2020)
[258]
P. D. Seymour, On incomparable collections of sets, Mathematika 20 (1973) pp. 208-209.
[259]
A. Shapira, Extremal Graph Theory, (Accessed on July 30, 2020)
[260]
A. Shapira, Topics in Extremal Combinatorics, (Accessed on July 30, 2020)
[261]
J. B. Shearer, A note on the independence number of triangle-free graphs, Discrete Math. 46 (1983) no. 1, pp. 83-87.
[262]
S. Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972) pp. 247-261.
[263]
A. Sidorenko, A correlation inequality for bipartite graphs, Graphs Combin. 9 (1993) no. 2, pp. 201-204.
[264]
A. F. Sidorenko, Inequalities for functionals generated by bipartite graphs, Diskret. Mat. 3 (1991) no. 3, pp. 50-65.
[265]
M. Simonovits, A method for solving extremal problems in graph theory, stability problems, (1968) pp. 279-319. Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic Press, New York;
[266]
M. Simonovits, Extremal graph problems, degenerate extremal problems, and supersaturated graphs, (1984) pp. 419-437. Progress in graph theory (Waterloo, Ont., 1982) Academic Press, Toronto, ON;
[267]
M. Simonovits, Paul Erdős’ influence on extremal graph theory, 14 (1997) pp. 148-192. The mathematics of Paul Erdős, II Algorithms Combin. Springer, Berlin;
[268]
O. Sisask, MTH711U: Extremal Combinatorics, 2011, Queen Mary, University of London, (Accessed on August, 2022)
[269]
J. Solymosi, A note on a question of Erdős and Graham, Combin. Probab. Comput. 13 (2004) no. 2, pp. 263-267.
[270]
J. Solymosi, Note on a generalization of Roth’s theorem, 25 (2003) pp. 825-827. Discrete and computational geometry Algorithms Combin. Springer, Berlin;
[271]
E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1928) no. 1, pp. 544-548.
[272]
J. M. Steele, The Cauchy-Schwarz master class, (2004) An introduction to the art of mathematical inequalities MAA Problem Books Series Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge;
[273]
B. Sudakov, LARGE INDEPENDENT SETS FROM LOCAL CONSIDERATIONS, (Viewed online on September 23, 2020) Probabilistic Combinatorics Online 2020 at MIPT,
[274]
B. Sudakov and J. Verstraëte, The extremal function for cycles of length \(\ell \bmod k\), Electron. J. Combin. 24 (2017) no. 1, Paper 1.7, 8.
[275]
B. Szegedy, An information theoretic approach to Sidorenko’s conjecture, (2015) arXiv:1406.6738v3
[276]
E. Szemerédi, On graphs containing no complete subgraph with \(4\) vertices, Mat. Lapok 23 (1972)
[277]
E. Szemerédi, On sets of integers containing no \(k\) elements in arithmetic progression, Acta Arith. 27 (1975) pp. 199-245.
[278]
E. Szemerédi, Regular partitions of graphs, 260 (1978) pp. 399-401. Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976) Colloq. Internat. CNRS CNRS, Paris;
[279]
T. Tao, Tricks Wiki article: the tensor power trick, (2008, Accessed on October 9, 2020)
[280]
T. Tao and V. Vu, The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and Füredi, Combinatorica 32 (2012) no. 3, pp. 363-372.
[281]
T. Tao and V. H. Vu, Inverse Littlewood-Offord theorems and the condition number of random discrete matrices, Ann. of Math. (2) 169 (2009) no. 2, pp. 595-632.
[282]
A. Thomason, An upper bound for some Ramsey numbers, J. Graph Theory 12 (1988) no. 4, pp. 509-517.
[283]
A. Thomason, Extremal Graph Theory, 2017, University of Cambridge. Notes scribed by Dexter Chua, (Accessed on July 30, 2020)
[284]
A. Thomason, Pseudorandom graphs, 144 (1987) pp. 307-331. Random graphs ’85 (Poznań, 1985) North-Holland Math. Stud. North-Holland, Amsterdam;
[285]
C. Thomassen, On the chromatic number of triangle-free graphs of large minimum degree, Combinatorica 22 (2002) no. 4, pp. 591-596.
[286]
I. Tomon, Extremal Combinatorics 2018, (Accessed on April 6, 2020)
[287]
P. Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941) pp. 436-452.
[288]
University of Cambridge, Part III of the Mathematical Tripos Examination Papers, (Accessed on October 27, 2022)
[289]
V. N. Vapnik and A. Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, pp. 11-30. Reprint of Theor. Probability Appl. \textbf16 (1971), 264--280 Measures of complexity Springer, Cham;
[290]
J. Verstraëte, Extremal problems for cycles in graphs, 159 (2016) pp. 83-116. Recent trends in combinatorics IMA Vol. Math. Appl. Springer, [Cham];
[291]
J. Verstraëte, On arithmetic progressions of cycle lengths in graphs, Combin. Probab. Comput. 9 (2000) no. 4, pp. 369-373.
[292]
B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw. Arch. Wisk. 15 (1927) pp. 212-216.
[293]
D. B. West, Combinatorial Mathematics, (2020) Cambridge University Press, Cambridge;
[294]
D. Williams, Probability with martingales, (1991) Cambridge Mathematical Textbooks Cambridge University Press, Cambridge;
[295]
K. Yamamoto, Logarithmic order of free distributive lattice, J. Math. Soc. Japan 6 (1954) pp. 343-353.
[296]
Y. Zhao, Extremal regular graphs: independent sets and graph homomorphisms, Amer. Math. Monthly 124 (2017) no. 9, pp. 827-843.
[297]
Y. Zhao, Graph Theory and Additive Combinatorics: Exploring Structure and Randomness, (Accessed on Dec 30, 2022)
[298]
Y. Zhao, The number of independent sets in a regular graph, Combin. Probab. Comput. 19 (2010) no. 2, pp. 315-320.
[299]
A. A. Zykov, On some properties of linear complexes, Mat. Sbornik N.S. 24(66) (1949) pp. 163-188.