The exact bound in the Erdös-Ko-Rado theorem

Wilson, R.

doi:10.1007/bf02579226

Cited by 258 publications

(200 citation statements)

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“…in [24] [5], [25] (and many others). Indeed, in [26] Wilson proves a result that is intimately related to ours, and a close comparison of our proof with his reveals many similarities. Another closely related work, whose method we generalize, is [21] which we also use as a didactic building block.…”

Section: History and Philosophysupporting

confidence: 78%

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On the measure of intersecting families, uniqueness and stability

Friedgut

2008

Combinatorica

126

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Let t ≥ 1 be an integer and let A be a family of subsets of {1, 2, . . . n} every two of which intersect in at least t elements. Identifying the sets with their characteristic vectors in {0, 1} n we study the maximal measure of such a family under a non uniform product measure. We prove, for a certain range of parameters, that the t-intersecting families of maximal measure are the families of all sets containing t fixed elements, and that the extremal examples are not only unique, but also stable: any t-intersecting family that is close to attaining the maximal measure must in fact be close in structure to a genuine maximum family. This is stated precisely in Theorem 1.6.We deduce some similar results for the more classical case of Erdős-Ko-Rado type theorems where all the sets in the family are restricted to be of a fixed size. See Corollary 1.7.The main technique that we apply is spectral analysis of intersection matrices that encode the relevant combinatorial information concerning intersecting families. An interesting twist is that part of the linear algebra involved is done over certain polynomial rings and not in the traditional setting over the reals.A crucial tool that we use is a recent result of Kindler and Safra [22] concerning Boolean functions whose Fourier transforms are concentrated on small sets.

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Section: History and Philosophysupporting

confidence: 78%

“…In [12] Frankl proved that for t ≥ 15 and for this range of k the maximum is indeed attained by principal families, and this was later extended to all values of t by Wilson in [26]. Frankl and Füredi made further progress towards settling the general question in [14].…”

Section: T-intersecting Families For T >mentioning

confidence: 99%

On the measure of intersecting families, uniqueness and stability

Friedgut

2008

Combinatorica

126

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“…Partial versions of this conjecture were proved in [Fra78], [FF91], [Wil84]. Fortunately, the complete intersection theorem for finite sets was settled not long ago by Ahlswede and Khachatrian [AK97].…”

Section: Analysis Of Boolean Functionsmentioning

confidence: 99%

On the hardness of approximating vertex cover

2005

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“…It is known that for n ≥ (w − t + 1)(t + 1), M (n, w, t) = n−t w−t [13], [14]. However, this is not the case for lower values of n. …”

Section: Lemma 12mentioning

confidence: 99%

Bounds on Codes Based on Graph Theory

Rouayheb

Georghiades

Soljanin³

et al. 2007

2007 IEEE International Symposium on Information Theory

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Abstract-Let Aq(n, d) be the maximum order (maximum number of codewords) of a q-ary code of length n and Hamming distance at least d. And let A(n, d, w) that of a binary code of constant weight w. Building on results from algebraic graph theory and Erdős-ko-Rado like theorems in extremal combinatorics, we show how several known bounds on Aq(n, d) and A(n, d, w) can be easily obtained in a single framework. For instance, both the Hamming and Singleton bounds can derived as an application of a property relating the clique number and the independence number of vertex transitive graphs. Using the same techniques, we also derive some new bounds and present some additional applications.

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The exact bound in the Erdös-Ko-Rado theorem

Cited by 258 publications

References 3 publications

On the measure of intersecting families, uniqueness and stability

On the measure of intersecting families, uniqueness and stability

On the hardness of approximating vertex cover

Bounds on Codes Based on Graph Theory

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