A new generalization of the Erdős-Ko-Rado theorem

Journal of Combinatorial Theory, Series A, 1986

Combinatorica, 2015

Paul Erdős and László Lovász proved in a landmark article that, for any positive integer k, up to isomorphism there are only finitely many maximal intersecting families of k−sets (maximal k−cliques). So they posed the problem of determining or estimating the largest number N (k) of the points in such a family. They also proved by means of an example that N (k) ≥ 2k − 2 + 1 2 2k−2 k−1. Much later, Zsolt Tuza proved that the bound is best possible up to a multiplicative constant by showing that asymptotically N (k) is at most 4 times this lower bound. In this paper we reduce the gap between the lower and upper bound by showing that asymptotically N (k) is at most 3 times the Erdős-Lovász lower bound. Conjecturally, the explicit upper bound obtained in this paper is only double the lower bound.

2019

For an integer d ≥ 2, a family F of sets is d-wise intersecting if for any distinct sets A1, A2, . . . , Ad ∈ F , A1 ∩ A2 ∩ · · · ∩ Ad 6= ∅, and non-trivial if ⋂ F = ∅. Hilton and Milner conjectured that for k ≥ d ≥ 2 and large enough n, the extremal non-trivial d-wise intersecting family of k-element subsets of [n] is one of the following two families: H(k, d) = {A ∈ ( [n] k ) : [d− 1] ⊂ A,A ∩ [d, k + 1] 6= ∅} ∪ {[k + 1] \ {i} : i ∈ [d− 1]} A(k, d) = {A ∈ ( [n] k ) : |A ∩ [d+ 1]| ≥ d}. The celebrated Hilton-Milner Theorem states that H(k, 2) is the unique extremal non-trivial intersecting family for k > 3. We prove the conjecture and prove a stability theorem, stating that any large enough non-trivial d-wise intersecting family of k-element subsets of [n] is a subfamily of A(k, d) or H(k, d).

Paul Erdős and László Lovász established that any maximal intersecting family of k−sets has at most k k blocks. They introduced the problem of finding the maximum possible number of blocks in such a family. They also showed that there exists a maximal intersecting family of k−sets with approximately (e − 1)k! blocks. Later Péter Frankl, Katsuhiro Ota and Norihide Tokushige used a remarkable construction to prove the existence of a maximal intersecting family of k−sets with at least (k 2) k−1 blocks. In this article we introduce the notion of a closed intersecting family of k−sets and show that such a family can always be embedded in a maximal intersecting family of k−sets. Using this result we present two examples which disprove two special cases of one of the conjectures of Frankl et al. This article also provides comparatively simpler construction of maximal intersecting families of k−sets with at least (k 2) k−1 blocks.

Archiv der Mathematik, 1987

Arxiv preprint arXiv:0808.0774, 2008

In this case, there is a plethora of non-intersecting families of k-sets. However, a bit of inspection reveals that there is an easy-to-define intersecting family that is quite populous: in particular, take an arbitrary element of [n] (say, 1) and consider the family of all sets containing that ...

Graphs and Combinatorics, 2006

In [21], Frank Plumpton Ramsey proved what has turned out to be a remarkable and important theorem which is now known as Ramsey's theorem. This result is a generalization of the pigeonhole principle and can now be seen as part of a family of theorems of the same flavour. These Ramsey-type theorems all have the common feature that they assert, in some precise combinatorial way, that if we deal with large enough sets of numhers, there will be some well behaved fragment in the set. In Harrington's words, Ramsey-type theorems assert that complete chaos is impossible. Ramsey-type theorems have turned out to be very important in a number of branches of mathematics. In this paper we shall survey a number of basic Ramsey-type theorems, and we will then look at a selection of applications of Ramsey-type theorems and Ramsey-type ideas. In the applications we will concentrate on graph theory, logic and complexity theory. Proofs will mostly not be given in detail, but it is hoped that the reader will gain some appreciation of the usefulness and importance of the beautiful area of asymptotic combinatorics.

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Resonance, 1997