Problem 813

Erdős Problem #813: declared status 'open'. Formalized: no. Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a clique on at least $h(n)$ vertices. Estimate $h(n)$ - in particular, do there exist constants $c_1,c_2>0$ such that$$n^{1/3+c_1}\ll h(n) \ll n^{1/2-c_2}?$$ Prize: no. Tags: graph theory.