OPEN
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 $1\leq q\leq \binom{p-1}{2}+1$?
A problem of Erdős, Faudree, Rousseau, and Schelp.
When $q=1$ this corresponds exactly to the classical Ramsey problem, and hence for example
\[\frac{1}{p-1}\leq c(p,1) \leq \frac{2}{p+1}.\]
It is easy to see that if $q=\binom{p-1}{2}+1$ then $c(p,q)=1$. Erdős, Faudree, Rousseau, and Schelp have shown that $c(p,\binom{p-1}{2})\leq 1/2$.
