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  and independently Keevash-Sudakov  settled an old conjecture of Sós  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
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.
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Research supported in part by NSF grants DMS-0653946 and DMS-0969092
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Mubayi, D. Counting substructures II: Hypergraphs. Combinatorica 33, 591–612 (2013). https://doi.org/10.1007/s00493-013-2638-2
Mathematics Subject Classification (2000)