Counting substructures II: Hypergraphs

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 KeevashSudakov [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) = (dn/2e 2 ) bn/2c + (bn/2c 2 ) dn/2e. We prove that there is an absolute constant c such that if n is sufficiently large and 1 ≤ q ≤ cn2, then every n vertex triple system with p(n) + q edges contains at least 6q (( bn/2c 4 ) + (dn/2e − 3) ( bn/2c 3 )) 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. ∗Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607. email: [email protected]; research supported in part by NSF grants DMS-0653946 and DMS-0969092 2000 Mathematics Subject Classification: 05A16, 05B07, 05D05