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Beyond the Erdős-Ko-Rado theorem

Peter Frankl, Zoltán Füredi
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  ... 
doi:10.1016/0097-3165(91)90031-b fatcat:v2wf6iyhv5coviogox43myfzsy

A simple removal lemma for large nearly-intersecting families

Tuan Tran, Shagnik Das
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.  ... 
doi:10.1016/j.endm.2015.06.015 fatcat:ru3yv5pg25hgxbzssl6lcqalcm

Erdős–Ko–Rado Theorem for a Restricted Universe

Peter Frankl
2020 Electronic Journal of Combinatorics  
In 1961 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|.  ... 
doi:10.37236/8682 fatcat:ldly3ygrh5dchbdlcnudswqybm

Removal and Stability for Erdős-Ko-Rado [article]

Shagnik Das, Tuan Tran
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.  ... 
arXiv:1412.7885v3 fatcat:vzugstetpjf3tfnrdlzlyt7mpu

Removal and Stability for Erdös--Ko--Rado

Shagnik Das, Tuan Tran
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.  ... 
doi:10.1137/15m105149x fatcat:a3zc4gaxejf6fa42f7tzulmdga

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

Karen Meagher, Lucia Moura
2005 Electronic Journal of Combinatorics  
In this paper, we prove a higher order generalization of 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.  ... 
doi:10.37236/1937 fatcat:sinxftgczrajzkyqvz4sjyouny

Beyond the Erdős Matching Conjecture [article]

Peter Frankl, Andrey Kupavskii
2020 arXiv   pre-print
In particular, we generalize 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.  ... 
arXiv:1901.09278v4 fatcat:5qxye42gxfgg3a4y6q44poai5i

Very Well-Covered Graphs with the Erdős-Ko-Rado Property [article]

Jessica De Silva, Adam B. Dionne, Aidan Dunkelberg, Pamela E. Harris
2022 arXiv   pre-print
A graph is r-EKR if 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] ).  ... 
arXiv:2106.09067v2 fatcat:l4in266cmjhmfbx33so5os2cqq

Improved bounds for Erdős' Matching Conjecture

Peter Frankl
2013 Journal of combinatorial theory. Series A  
More than 50 years ago, 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.  ... 
doi:10.1016/j.jcta.2013.01.008 fatcat:i67haze77ndfpm6eebvzetg4z4

Intersecting families of discrete structures are typically trivial

József Balogh, Shagnik Das, Michelle Delcourt, Hong Liu, Maryam Sharifzadeh
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.  ... 
doi:10.1016/j.jcta.2015.01.003 fatcat:rjn3jvahcjg5ndodg6wwh2n5tu

Intersecting families of discrete structures are typically trivial [article]

József Balogh, Shagnik Das, Michelle Delcourt, Hong Liu, Maryam Sharifzadeh
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.  ... 
arXiv:1408.2559v4 fatcat:wxdzezhrc5gvfg5ptg4fk5mr64

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

Ali Taherkhani
2022 Electronic Journal of Combinatorics  
.$ 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).  ... 
doi:10.37236/10490 fatcat:wswu4qabovbdtol3xci34chg3i

The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton–Milner family

Jie Han, Yoshiharu Kohayakawa
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.  ... 
doi:10.1090/proc/13221 fatcat:j7um4u7jn5fpxipf7cqkg5o7lu

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

Ali Taherkhani
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).  ... 
arXiv:1903.02614v3 fatcat:eso6mul4wzg4xkd752o6oyvvvu

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

Karen Meagher
2007 arXiv   pre-print
It is known that 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 . . . . . . . . .  ... 
arXiv:math/0701553v1 fatcat:cgo6qolrang5vkxqkyvvesfqly
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