Given sets \(X_1\) and \(X_2\text{,}\) say that \(X_1\) is homothetic to \(X_2\) if there exist \(\vec{p}\in\mathbb{R}^d\) and \(r>0\) such that \(X_2=\vec{p}+rX_1\text{.}\) Say that a set \(A\subseteq \mathbb{R}^d\) is \(X\)-free if it does not contain any subset that is homothetic to \(X\text{.}\) Using Behrend’s Theorem (see Exercise 10 from Section 5.4), prove that for any set \(X\subseteq\mathbb{R}^d\) with \(|X|\geq3\text{,}\) there exists \(c>0\) such that, for every \(n\geq1\text{,}\) there exists an \(X\)-free set \(A\subseteq [n]^d\) satisfying
Recall that \(\alpha(G)\) is the cardinality of the largest independent set in \(G\) (i.e. the largest set of vertices containing no edges). Show that if \(G\) is a \(K_4\)-free graph with \(n\) vertices satisfying \(\alpha(G)=o(n)\text{,}\) then