An Extension Of The Ruzsa-Szemerédi Theorem

We let G ^(r)(n,m) denote the set of r-uniform hypergraphs with n vertices and m edges, and f ^(r)(n,p,s) is the smallest m such that every member of G ^(r)(n,m) contains amember of G ^(r)(p,s). In this paper we are interested in fixed values r,p and s for which f ^(r)(n,p,s) grows quadratically with n. A probabilistic construction of Brown, Erdős and T. Sós ([2]) implies that f ^(r)(n,s(r-2)+2,s)=Ω(n ²). In the other direction the most interesting question they could not settle was whether f ⁽³⁾(n,6, 3) = o(n ²). This was proved by Ruzsa and Szemerédi [11]. Then Erdős, Frankl and Rödl [6] extended this result to any r: f ^(r)(n, 3(r-2)+3, 3)=o(n ²), and they conjectured ([4], [6]) that the Brown, Erdős and T. Sós bound is best possible in the sense that f ^(r)(n,s(r-2)+3,s)=o(n ²).

In this paper by giving an extension of the Erdős, Frankl, Rödl Theorem (and thus the Ruzsa–Szemerédi Theorem) we show that indeed the Brown, Erdős, T. Sós Theorem is not far from being best possible. Our main result is

$$ f^{{{\left( r \right)}}} {\left( {n,s{\left( {r - 2} \right)} + 2 + {\left\lfloor {\log _{2} s} \right\rfloor },s} \right)} = o{\left( {n^{2} } \right)}. $$