Colour each edge of the complete graph on \(n\) vertices red with probability \(p\) and blue with probability \(1-p\text{,}\) independently of one another. What is the expected number of red copies of \(K_r\text{?}\) The expected number of blue copies of \(K_s\text{?}\)
Let \(G\) be a bipartite graph with \(n\) vertices. Suppose that each vertex of \(G\) is assigned a list \(L(v)\) of more than \(\log_2(n)\) colours. Show that there is a proper colouring of \(G\) in which, for every vertex \(v\text{,}\) the colour of \(v\) is contained in \(L(v)\text{.}\)
For \(n\geq r\geq1\text{,}\) let \(\mathcal{F}\subseteq \binom{[n]}{r}\text{.}\) Prove that it is possible to colour \([n]\) with \(r\) colours such that at least \(\frac{r!}{r^r}|\mathcal{F}|\) elements of \(\mathcal{F}\) contain a point of every colour.
Let \(\mathcal{F}\) be a finite collection of binary strings of finite length (but not necessarily the same length). For each string \(s\in\mathcal{F}\text{,}\) let \(\ell(s)\) denote the length of \(s\text{.}\) Prove that
