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Beyond the Erdős-Ko-Rado theorem

1991
*
Journal of combinatorial theory. Series A
*

*Erdos*,

*Ko*, and

*Rado*[EKR] proved that this condition implies 1.9 < (ix:) whenever n > n, (k, t) . ... P,, is

*the*convex hull of

*the*f(F(K, r))'s. Corollaries. As (k -t + 1 )(t + 1) = n,(k, t) d i(k + l)*, one can formulate

*the*exact version of

*the*Erdiis-

*Ko*-

*Rado*

*theorem*as follows. ... In

*the*last step we used (5.8). On

*the*other hand PI =2(,-:-*)= .",",:; 1 (kl,)+$(k:*). t5.18) Finally, (5.17), (5.18), and (5.1) give Claim 5.16. i ...

##
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A simple removal lemma for large nearly-intersecting families

2015
*
Electronic Notes in Discrete Mathematics
*

*The*

*Erdős*-

*Ko*-

*Rado*

*theorem*shows α(K(n, k)) = n−1 k−1 . ...

*The*study of intersecting families is central to extremal set theory, dating back to

*the*seminal

*Erdős*-

*Ko*-

*Rado*

*theorem*of 1961 that bounds

*the*largest such families. ... an extension of

*the*

*Erdős*-

*Ko*-

*Rado*

*theorem*to

*the*sparse random setting. ...

##
###
Erdős–Ko–Rado Theorem for a Restricted Universe

2020
*
Electronic Journal of Combinatorics
*

In 1961

doi:10.37236/8682
fatcat:ldly3ygrh5dchbdlcnudswqybm
*Erdős*,*Ko*and*Rado*showed that $|\mathcal F| \leq {n - 1\choose k - 1}$ if $n \geq 2k$. ...*The*paper of Li et al. is one of them. ...*The*classical*Erdős*-*Ko*-*Rado**Theorem*[EKR] states that no k-uniform intersecting family can surpass |S|. ...##
###
Removal and Stability for Erdős-Ko-Rado
[article]

2016
*
arXiv
*
pre-print

*The*

*Erdős*-

*Ko*-

*Rado*

*theorem*shows α(K(n,k)) = n-1k-1. ...

*The*study of intersecting families is central to extremal set theory, dating back to

*the*seminal

*Erdős*-

*Ko*-

*Rado*

*theorem*of 1961 that bounds

*the*size of

*the*largest such families. ... Acknowledgements We would like to thank Jozsef Balogh, Hong Liu and Maryam Sharifzadeh for suggesting

*the*generalisation of

*the*removal lemma to larger families, with ℓ ≥ 2. ...

##
###
Removal and Stability for Erdös--Ko--Rado

2016
*
SIAM Journal on Discrete Mathematics
*

*The*study of intersecting families is central to extremal set theory, dating back to

*the*seminal

*Erdős*-

*Ko*-

*Rado*

*theorem*of 1961 that bounds

*the*size of

*the*largest such families. ... For some constant c > 0 and k ≤ cn, we determine

*the*sharp threshold for when this equality holds for random subgraphs of K(n, k), and provide strong bounds on

*the*critical probability for k ≤ 1 2 (n − ... Acknowledgements We would like to thank Jozsef Balogh, Hong Liu and Maryam Sharifzadeh for suggesting

*the*generalisation of

*the*removal lemma to larger families, with ℓ ≥ 2. ...

##
###
Erdős-Ko-Rado theorems for uniform set-partition systems

2005
*
Electronic Journal of Combinatorics
*

In this paper, we prove a higher order generalization of

doi:10.37236/1937
fatcat:sinxftgczrajzkyqvz4sjyouny
*the**Erdős*-*Ko*-*Rado**theorem*for systems of pairwise $t$-intersecting uniform $k$-partitions of an $n$-set. ... A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has*the*same cardinality $c=n/k$. ...*Erdős*and Székely observe that*the*following*Erdős*-*Ko*-*Rado*type result for t-intersecting partition systems holds. ...##
###
Beyond the Erdős Matching Conjecture
[article]

2020
*
arXiv
*
pre-print

In particular, we generalize

arXiv:1901.09278v4
fatcat:5qxye42gxfgg3a4y6q44poai5i
*the*result of*the*first author on*the**Erdős*Matching Conjecture and prove a generalization of*the**Erdős*-*Ko*-*Rado**theorem*, which states that for n> s^2k*the*largest family ℱ⊂ ... We investigate*the*case k=3 more thoroughly, showing that, unlike in*the*case of*the**Erdős*Matching Conjecture, in general there may be 3 extremal families. ...*The*research of both authors was supported by*the*Ministry of Education and Science of*the*Russian Federation in*the*framework of MegaGrant no 075- 15-2019-1926. ...##
###
Very Well-Covered Graphs with the Erdős-Ko-Rado Property
[article]

2022
*
arXiv
*
pre-print

A graph is r-EKR if

arXiv:2106.09067v2
fatcat:l4in266cmjhmfbx33so5os2cqq
*the*maximum size of an intersecting family of independent r-sets is*the*size of an r-star. ... We prove that*the*pendant complete graph K_n^* is r-EKR when n ≥ 2r and strictly so when n>2r. Pendant path graphs P_n^* are also explored and*the*vertex whose r-star is of maximum size is determined. ... This naming stems from*the*classical*Erdös*-*Ko*-*Rado**theorem*, framed in*the*language of graph theory as follows:*Theorem*1 (*Erdös*-*Ko*-*Rado*[3] ). ...##
###
Improved bounds for Erdős' Matching Conjecture

2013
*
Journal of combinatorial theory. Series A
*

More than 50 years ago,

doi:10.1016/j.jcta.2013.01.008
fatcat:i67haze77ndfpm6eebvzetg4z4
*Erdős*asked*the*following question: what is*the*largest family of k-element subsets of [n] with no s pairwise disjoint sets? ...*The*case s = 2 is*the*classical*Erdős*-*Ko*-*Rado**theorem*[7] which was*the*starting point of a large part of ongoing research in extremal set theory. ...*Beyond**the**Erdős*Matching Conjecture Let us introduce*the*following general notion. Definition 1. Let k, s ≥ 2 and k ≤ q < sk be integers. ...##
###
Intersecting families of discrete structures are typically trivial

2015
*
Journal of combinatorial theory. Series A
*

*The*classic

*Erdős*-

*Ko*-

*Rado*

*theorem*shows that

*the*largest t-intersecting k-uniform hypergraphs are also trivial when n is large. ... Our proofs use

*the*Bollobás set-pairs inequality to bound

*the*number of maximal intersecting families, which can then be combined with known stability

*theorems*. ... Acknowledgement We would like to thank

*the*University of Szeged for their kind hospitality, and

*the*referees for their careful reading of this paper. ...

##
###
Intersecting families of discrete structures are typically trivial
[article]

2015
*
arXiv
*
pre-print

*The*classic

*Erdős*--

*Ko*--

*Rado*

*theorem*shows that

*the*largest t-intersecting k-uniform hypergraphs are also trivial when n is large. ... Our proofs use

*the*Bollobás set-pairs inequality to bound

*the*number of maximal intersecting families, which can then be combined with known stability

*theorems*. ... Acknowledgement We would like to thank

*the*University of Szeged for their kind hospitality, and

*the*referees for their careful reading of this paper. ...

##
###
Size and Structure of Large $(s,t)$-Union Intersecting Families

2022
*
Electronic Journal of Combinatorics
*

.$

doi:10.37236/10490
fatcat:wswu4qabovbdtol3xci34chg3i
*The*celebrated*Erdős*-*Ko*-*Rado**theorem*determines*the*size and structure of*the*largest intersecting (or $(1,1)$-union intersecting) family. ... Our results are nontrivial extensions of some recent generalizations of*the**Erdős*-*Ko*-*Rado**theorem*such as*the*Han and Kohayakawa*theorem*~[Proc. Amer. Math. ... Acknowledgements*The*author is grateful to Meysam Alishahi and Amir Daneshgar for their valuable comments. This research was in part supported by a grant from IPM (No. 98050012). ...##
###
The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton–Milner family

2016
*
Proceedings of the American Mathematical Society
*

*The*celebrated

*Erdős*-

*Ko*-

*Rado*

*theorem*determines

*the*maximum size of a k-uniform intersecting family. ...

*The*Hilton-Milner

*theorem*determines

*the*maximum size of a k-uniform intersecting family that is not a subfamily of

*the*so-called

*Erdős*-

*Ko*-

*Rado*family. ... Acknowledgement We thank an anonymous referee for many helpful comments that helped us improve

*the*presentation of

*the*paper. ...

##
###
Size and structure of large (s,t)-union intersecting families
[article]

2019
*
arXiv
*
pre-print

*The*celebrated

*Erdős*-

*Ko*-

*Rado*

*theorem*determines

*the*size and structure of

*the*largest intersecting family of k-sets on an n-set X. ... Our results are nontrivial extensions of some recent generalizations of

*the*

*Erdős*-

*Ko*-

*Rado*

*theorem*such as

*the*Han and Kohayakawa

*theorem*2017 which finds

*the*structure of

*the*third largest intersecting ... Acknowledgements

*The*author is grateful to Meysam Alishahi and Amir Daneshgar for their valuable comments. This research was in part supported by a grant from IPM (No. 98050012). ...

##
###
Covering arrays on graphs: qualitative independence graphs and extremal set partition theory
[article]

2007
*
arXiv
*
pre-print

It is known that

arXiv:math/0701553v1
fatcat:cgo6qolrang5vkxqkyvvesfqly
*the*exact size of an optimal binary covering array can be determined using Sperner's*Theorem*and*the**Erdos*-*Ko*-*Rado**Theorem*. ... Since*the*rows of general covering arrays correspond to set partitions, we give extensions of Sperner's*Theorem*and*the**Erdos*-*Ko*-*Rado**Theorem*to set-partition systems. ... 38 3.3.2*The**Erdős*-*Ko*-*Rado**Theorem*. . . . . . . . . . . . . . . . . . 39 Generalizations of*the**Erdős*-*Ko*-*Rado**Theorem*. . . . . . . . 39 3.4 Application of*the**Erdős*-*Ko*-*Rado**Theorem*. . . . . . . . . ...
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