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.