The structure of large intersecting families

A collection of sets is intersecting if every two members have nonempty intersection. We describe the structure of intersecting families of r-sets of an n-set whose size is quite a bit smaller than the maximum n 1 r 1 given by the Erd} os-Ko-Rado Theorem. In particular, this extends the Hilton-Milner theorem on nontrivial intersecting families and answers a recent question of Han and Kohayakawa for large n. In the case r = 3 we describe the structure of all intersecting families with more than 10 edges. We also prove a stability result for the Erd} os matching problem. Our short proofs are simple applications of the Delta-system method introduced and extensively used by Frankl since 1977.

[1]  P. Erdos-L Lovász.  Problems and Results on 3-chromatic Hypergraphs and Some Related Questions , 2022 .

[2]  Yoshiharu Kohayakawa,et al.  The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton--Milner family , 2015 .

[3]  Peter Frankl,et al.  On intersecting families of finite sets , 1978, Bulletin of the Australian Mathematical Society.

[4]  Zoltán Füredi,et al.  Linear trees in uniform hypergraphs , 2012, Eur. J. Comb..

[5]  TOMASZ LUCZAK,et al.  On Erdős' extremal problem on matchings in hypergraphs , 2012, J. Comb. Theory, Ser. A.

[6]  Zoltán Füredi,et al.  Hypergraph Turán numbers of linear cycles , 2014, J. Comb. Theory, Ser. A.

[7]  P. Erdös,et al.  INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1961 .

[8]  Peter Frankl,et al.  On the maximum number of edges in a hypergraph with given matching number , 2012, Discret. Appl. Math..

[9]  A. J. W. Hilton,et al.  SOME INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1967 .

[10]  D. E. Daykin,et al.  SETS OF INDEPENDENT EDGES OF A HYPERGRAPH , 1976 .

[11]  Peter Frankl,et al.  Improved bounds for Erdős' Matching Conjecture , 2013, J. Comb. Theory, Ser. A.

[12]  A. Tsarpalias,et al.  A Combinatorial Theorem , 1981, J. Comb. Theory, Ser. A.

[13]  Hao Huang,et al.  The Size of a Hypergraph and its Matching Number , 2011, Combinatorics, Probability and Computing.

[14]  Peter Frankl,et al.  Improved bounds for Erd ˝ os' Matching Conjecture , 2013 .

[15]  Peter Frankl.  On families of finite sets no two of which intersect in a singleton , 1977, Bulletin of the Australian Mathematical Society.

[16]  Peter Frankl,et al.  Erdös-Ko-Rado theorem with conditions on the maximal degree , 1987, J. Comb. Theory, Ser. A.

[17]  P. Erdos.  A PROBLEM ON INDEPENDENT r-TUPLES , 1965 .