Counting substructures II: Hypergraphs

Abstract

For various k-uniform hypergraphs F, we give tight lower bounds on the number of copies of F in a k-uniform hypergraph with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of various authors who proved that there is one copy of F.

A sample result is the following: Füredi-Simonovits [11] and independently Keevash-Sudakov [16] settled an old conjecture of Sós [29] by proving that the maximum number of triples in an n vertex triple system (for n sufficiently large) that contains no copy of the Fano plane is p(n)=( ^⌈n/2⌉₂ )⌊n/2⌋+( ^⌊n/2⌋₂ ⌈n/2⌉). We prove that there is an absolute constant c such that if n is sufficiently large and 1 ≤ q ≤ cn ², then every n vertex triple system with p(n)+q edges contains at least

$6q\left( {\left( {_4^{\left\lfloor {n/2} \right\rfloor } } \right) + \left( {\left\lceil {n/2} \right\rceil - 3} \right)\left( {_3^{\left\lfloor {n/2} \right\rfloor } } \right)} \right)$

copies of the Fano plane. This is sharp for q≤n/2–2.

Our proofs use the recently proved hypergraph removal lemma and stability results for the corresponding Turán problem.