On A Hypergraph Turán Problem Of Frankl

Let $C^{{{\left( {2k} \right)}}}_{r}$ be the 2k-uniform hypergraph obtained by letting P₁, . . .,P_r be pairwise disjoint sets of size k and taking as edges all sets P _i ∪P _j with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph K _r : 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 2^p+1. In this paper we also show that when r=2^p+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 Turán 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.