On A Hypergraph Turán Problem Of Frankl

Abstract

Let C (2k) r be the 2k-uniform hypergraph obtained by letting P 1 , · · · , 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 (2k) 3 can be obtained by partitioning V = V 1 ∪V 2 and taking as edges all sets of size 2k that intersect each of V 1 and V 2 in an odd number of elements. Let B (2k) n denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. We prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no C (2k) 3 has at most as many edges as B (2k) n . Sidorenko has given an upper bound of r−2 r−1 for the Turán density of C (2k) 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 (4) r -free hypergraph of density r−2 r−1 − o(1) 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 (4) r to r−2 r−1 − c(r), 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.

An exact Turán result for the generalized triangle

“…Let us remark here that the fact that stability may help in exact computation of the hypergraph Turán function was observed and used by Füredi and Simonovits [7], Keevash and Mubayi [11], Keevash and Sudakov [13,12], Mubayi and Pikhurko [15], Pikhurko [17], and others.…”

Set systems without a simplex or a cluster

“…Indeed, it was developed by Erdős and Simonovits to determine the exact Turán number for k-critical graphs. This approach has been recently used with great success in hypergraph Turán theory (see [18,19,20,24,25,29,32,33]), enumeration of discrete structures [4] and extremal set theory (see [2,26,27]). …”

The Turán problem for projective geometries

“…Most of these are described in the excellent survey of Füredi [4]. More recently, there have been three new exact results (see [6][7][8]11,12] for details). Most of the progress has been for triple systems (the case r = 3), where there have also been some new results on Turán densities (see [14]).…”