The maximum size of 3-uniform hypergraphs not containing a fano plane
Journal of Combinatorial Theory Series B
On the Turán number of triple systems
Journal of Combinatorial Theory Series A
Stability theorems for cancellative hypergraphs
Journal of Combinatorial Theory Series B
Triple Systems Not Containing a Fano Configuration
Combinatorics, Probability and Computing
On Triple Systems with Independent Neighbourhoods
Combinatorics, Probability and Computing
Applications of the regularity lemma for uniform hypergraphs
Random Structures & Algorithms
The counting lemma for regular k-uniform hypergraphs
Random Structures & Algorithms
The Turán Number Of The Fano Plane
Combinatorica
A hypergraph extension of Turán's theorem
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series A
On A Hypergraph Turán Problem Of Frankl
Combinatorica
A variant of the hypergraph removal lemma
Journal of Combinatorial Theory Series A
A new generalization of Mantel's theorem to k-graphs
Journal of Combinatorial Theory Series B
An exact Turán result for the generalized triangle
Combinatorica
On the minimal density of triangles in graphs
Combinatorics, Probability and Computing
Note: Quadruple systems with independent neighborhoods
Journal of Combinatorial Theory Series A
Exact computation of the hypergraph Turán function for expanded complete 2-graphs
Journal of Combinatorial Theory Series B
Hi-index | 0.00 |
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)=( 2 驴n/2驴 )驴n/2驴+( 2 驴n/2驴 驴n/2驴). 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.