A hypergraph regularity method for generalized Turán problems

Abstract

ABSTRACT:We describe a method that we believe may be foundational for a comprehensive theory of generalized Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that is, any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2 − q + 1 and n/2, but for PG 2 (4) we find the somewhat surprising phenomenon that the threshold is less than (1/2 − )n for some small > 0. We conclude by setting out a program for future developments of this method to tackle other problems.

Hypergraphs with independent neighborhoods

“…[11,Corollary 2.3].) Therefore, the size of the family F is at least (1 − 1/l)(1 − 1/k 2 ) n k , and (16) • A related problem which has been studied a fair amount recently (see, e.g., [13,15,17,22]) is the maximum possible minimum degree (of (k − 1)-sets) that a k-graph can have without containing some fixed configuration. Let g(n, k) denote the maximum minimum degree of a k-graph on n vertices with independent neighborhoods.…”

Counting designs

“…It is well-known that G is c 1/50 -regular (say) in the 'Szemerédi sense' (see e.g. [6,Theorem 2.2]). For S ⊆ V (G) with |S| ≤ 16, say that S is good if | ∩ x∈S G(x)| = (1 ± |S|c 1/100 )d(G) |S| n, otherwise bad.…”

Codegree Thresholds for Covering 3-Uniform Hypergraphs

“…Keevash and Zhao [18] determined the codegree densities of some projective geometries, which included the Fano plane as a special case. The codegree threshold for the Fano plane was determined by Keevash [15] via hypergraph regularity and later by DeBiasio and Jiang [5] by direct combinatorial means. Mubayi and Zhao [27] studied general properties of codegree density, while Falgas-Ravry [6] gave examples of non-isomorphic lower bound constructions for γ(K t ).…”