On a Tur\'an problem in weakly quasirandom 3-uniform hypergraphs

Extremal problems for $3$-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is weakly $(d,\eta)$-quasirandom if for any subset $U\subseteq V$ the number of hyperedges of $H$ contained in $U$ is in the interval $d\binom{|U|}{3}\pm\eta|V|^3$. We show that for any $\varepsilon>0$ there exists $\eta>0$ such that every sufficiently large weakly $(1/4+\varepsilon,\eta)$-quasirandom hypergraph contains four vertices spanning at least three hyperedges. This was conjectured by Erd\H{o}s and S\'os and it is known that the density $1/4$ is best possible. Recently, a computer assisted proof of this result based on the flag-algebra method was established by Glebov, Kr\'al', and Volec. In contrast to their work our proof presented here is based on the regularity method of hypergraphs and requires no heavy computations. In addition we obtain an ordered version of this result. The method of our proof allows us to study extremal problems of this type in a more systematic way and we discuss a few extensions and open problems here.