On A Hypergraph Turán Problem Of Frankl

Abstract

Let $$C^{{{\left( {2k} \right)}}}_{r}$$ be the 2k-uniform hypergraph obtained by letting P1, . . .,Pr be pairwise disjoint sets of size k and taking as edges all sets Pi∪Pj with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph Kr: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain $$C^{{{\left( {2k} \right)}}}_{3}$$ can be obtained by partitioning V = V1 ∪V2 and taking as edges all sets of size 2k that intersect each of V1 and V2 in an odd number of elements. Let $${\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n}$$ denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no $$C^{{{\left( {2k} \right)}}}_{3}$$ has at most as many edges as $${\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n}$$.Sidorenko has given an upper bound of $$\frac{{r - 2}}{{r - 1}}$$ for the Tur´an density of $$C^{{{\left( {2k} \right)}}}_{r}$$ for any r, and a construction establishing a matching lower bound when r is of the form 2p+1. In this paper we also show that when r=2p+1, any $$C^{{{\left( 4 \right)}}}_{r}$$-free hypergraph of density $$\frac{{r - 2}}{{r - 1}} - o{\left( 1 \right)}$$ looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turan density of $$C^{{{\left( 4 \right)}}}_{r}$$ to $$\frac{{r - 2}}{{r - 1}} - c{\left( r \right)}$$, where c(r) is a constant depending only on r.The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials.