Problem 667

Erdős Problem #667: declared status 'open'. Formalized: no. Let $p,q\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices. Is$$c(p,q)=\liminf \frac{\log H(n)}{\log n}$$a strictly increasing function of $q$ for $1\leq q\leq \binom{p-1}{2}+1$? Prize: no. Tags: graph theory, ramsey theory.