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An Erdős-Ko-Rado Theorem for Multisets

Karen Meagher, Alison Purdy
2011 Electronic Journal of Combinatorics  
The size and structure of the largest intersecting collection of $k$-multisets for $m \leq k$ is also given.  ...  We prove that for $m \geq k+1$, the size of the largest such collection is $\binom{m+k-2}{k-1}$ and that when $m > k+1$, only a collection of all the $k$-multisets containing a fixed element will attain  ...  Acknowledgment We are grateful for the helpful comments of the anonymous referee, particularly those concerning Theorem 1.3 which greatly simplified the proof.  ... 
doi:10.37236/707 fatcat:k4dcwq6aiva3rjviwucpnmbcxy

An Erdős-Ko-Rado theorem for multisets [article]

Karen Meagher, Alison Purdy
2011 arXiv   pre-print
The size and structure of the largest intersecting collection of k-multisets for m ≤ k is also given.  ...  We prove that for m ≥ k+1, the size of the largest such collection is m+k-2k-1 and that when m > k+1, only a collection of all the k-multisets containing a fixed element will attain this bound.  ...  Acknowledgment We are grateful for the helpful comments of the anonymous referee, particularly those concerning Theorem 1.3 which greatly simplified the proof.  ... 
arXiv:1111.4493v1 fatcat:d6pqiyz6vvbyncguarlqyx7oce

An Erdös-Ko-Rado theorem for multisets

Ian Anderson
1988 Discrete Mathematics  
vector in SC An Erdiis-Ko-Rado rAeorem for multisets &$. are in 9i U l l l U !  ...  In looking for extensions of the Erdiis-Ko-Rado theorem to multisets, we observe that we can extend the concept of intersection in more than one way.  ... 
doi:10.1016/0012-365x(88)90172-0 fatcat:foxnpc6hxbhwpbjwh5j2lea5mq

Elementary Techniques for Erdos-Ko-Rado-like Theorems [article]

Greg Brockman, Bill Kay
2008 arXiv   pre-print
The well-known Erdos-Ko-Rado Theorem states that if F is a family of k-element subsets of 1,2,...  ...  We present elementary methods for deriving generalizations of the Erdos-Ko-Rado Theorem on several classes of combinatorial objects. We also extend our results to systems under Hamming intersection.  ...  Anant Godbole for his supervision at the 2008 East Tennessee State University REU. This work was supported by NSF grant 0552730.  ... 
arXiv:0808.0774v2 fatcat:hsoy3xfoqvgevgxfz2zbtjakc4

A discrete isodiametric result: the Erdős-Ko-Rado theorem for multisets [article]

Zoltán Füredi, Dániel Gerbner, Máté Vizer
2014 arXiv   pre-print
There are many generalizations of the Erdős-Ko-Rado theorem.  ...  We give new results (and problems) concerning families of t-intersecting k-element multisets of an n-set and point out connections to coding theory and classical geometry.  ...  Erdős-Ko-Rado type theorems Let us call a set system F intersecting if |F 1 ∩ F 2 | ≥ 1 for all F 1 , F 2 ∈ F.  ... 
arXiv:1212.1071v2 fatcat:wufurhu7jne4ziv3jwbt2mkyw4

A discrete isodiametric result: The Erdős–Ko–Rado theorem for multisets

Zoltán Füredi, Dániel Gerbner, Máté Vizer
2015 European journal of combinatorics (Print)  
There are many generalizations of the Erdős-Ko-Rado theorem.  ...  We give new results (and problems) concerning families of t-intersecting k-element multisets of an n-set and point out connections to coding theory and classical geometry.  ...  Erdős-Ko-Rado type theorems Let us call a set system F intersecting if |F 1 ∩ F 2 | ≥ 1 for all F 1 , F 2 ∈ F.  ... 
doi:10.1016/j.ejc.2015.02.023 fatcat:rozxnnj5jzaqlmqaazcx4yoa2q

Intersection theorems for multisets [article]

Karen Meagher, Alison Purdy
2015 arXiv   pre-print
These results include a multiset version of the Hilton-Milner theorem and a theorem giving the size and structure of the largest t-intersecting family of k-multisets of an m-set when m ≤ 2k-t.  ...  We use graph homomorphisms and existing theorems for intersecting and t-intersecting k-set systems to prove new results for intersecting and t-intersecting families of k-multisets.  ...  The second author was supported by an NSERC doctoral scholarship. We would like to thank Zoltan Füredi for bringing reference [7] to our attention.  ... 
arXiv:1504.06657v2 fatcat:ikbzeewhpnhbhd5wyyeaomkaa4

Erdős-Ko-Rado-type results over Jq(n, d), Hq(n, d) and their designs

Tung-Shan Fu
1999 Discrete Mathematics  
In terms of the notion of a specific class of ranked semilattices, called regular quantum matroids, we prove the Erdiis-Kc-Rado-type results in a unified way for the association schemes J,(n,d) of vector  ...  Huang for introducing him to the Erdiis-Ko-Rado type results and for pointing out the arguments used in [6] . The author would also like to thank both referees for their comments.  ...  On the other hand, an extension of Erdos-K+Rado theorem still holds if F is replaced by a collection of blocks of classical t-(n, k, A) designs.  ... 
doi:10.1016/s0012-365x(98)00171-x fatcat:prb7nvgutfb4fo6dfgqtas6qpm

The Spectral Radii of Intersecting Uniform Hypergraphs

Peng-Li Zhang, Xiao-Dong Zhang
2020 Communications on Applied Mathematics and Computation  
This paper states spectral versions of the Erdős-Ko-Rado theorem: let G be an intersecting k-uniform hypergraph on n vertices with n ⩾ 2k.  ...  The celebrated Erdős-Ko-Rado theorem states that given n ⩾ 2k, every intersecting k-uniform hypergraph G on n vertices has at most n − 1 k − 1 edges.  ...  Acknowledgements The authors would like to thank the anonymous referees for their careful reading and providing helpful suggestions and comments on an earlier version of this paper, which lead to a great  ... 
doi:10.1007/s42967-020-00073-7 fatcat:qizeyvoywvazpm7wzvrgqlflva

Pseudo-LYM Inequalities and AZ Identities

Péter L. Erdős, László A. Székely
1997 Advances in Applied Mathematics  
We give pseudo-LYM inequalities in some posets and give a new restriction in this way for their antichains. Typically these posets fail the LYM inequality and some of them are known not to be Sperner.  ...  Erdős, Seress, and Székely proved an Erdős-Ko-Rado theorem for the chain poset ( [10] ).  ...  Observe that the number of choices for the coarser partition is exactly N (γ). 2 Observe that Theorem 2.2 implies Theorem 2.1.  ... 
doi:10.1006/aama.1997.0542 fatcat:yf6b32bcjjhlbfkn7g2h5knqmm

Erdős-Ko-Rado theorem and bilinear forms graphs for matrices over residue class rings [article]

Jun Guo
2020 arXiv   pre-print
As a result, the Erdős-Ko-Rado theorem for Z_h^m× n is obtained.  ...  The results on Erdős-Ko-Rado theorem have inspired much research [6, 8, 17] . Let 1 ≤ r ≤ m ≤ n.  ...  As a result, the Erdős-Ko-Rado theorem for Z m×n h is obtained. Suppose that α i ≥ s i for each i ∈ I, and 0 ≤ α j < s j for each j ∈ [t] \ I.  ... 
arXiv:2002.03560v1 fatcat:765th7chwfczpf2mq2duzhpjsu

High dimensional Hoffman bound and applications in extremal combinatorics

Yuval Filmus, Konstantin Golubev, Noam Lifshitz
2022 Algebraic Combinatorics  
One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues.  ...  The n-th tensor power of a graph with vertex set V is the graph on the vertex set V n , where two vertices are connected by an edge if they are connected in each coordinate.  ...  The authors are grateful to Ehud Friedgut, Gil Kalai, Guy Kindler, and Dor Minzer for valuable discussions. We thank the reviewers for several helpful comments and suggestions.  ... 
doi:10.5802/alco.190 fatcat:fghev5ns45c2pahyk2a2ncqj4a

High dimensional Hoffman bound and applications in extremal combinatorics [article]

Yuval Filmus, Konstantin Golubev, Noam Lifshitz
2019 arXiv   pre-print
As another application, we provide spectral proofs for Mantel's theorem on triangle-free graphs and for Frankl-Tokushige theorem on k-wise intersecting families.  ...  One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues.  ...  Acknowledgements The authors are grateful to Ehud Friedgut, Gil Kalai, Guy Kindler, and Dor Minzer for valuable discussions. Funding.  ... 
arXiv:1911.02297v1 fatcat:zf7lwyvnw5bldlw4zahqmy7thi

Advances on Extremal Problems in Number Theory and Combinatorics [chapter]

Rudolf Ahlswede
2001 European Congress of Mathematics  
It implies also an Intersection Theorem for multisets of Erdős and Schönheim from 1969.  ...  Erdős, Seress, Székely [ErSS] and Füredi concerning an Erdős-Ko-Rado-type intersection property for the poset of Boolean chains could also be established.  ...  We also present an extension to multi-sets and explain a connection to the (higher dimensional) Erdős-Moser problem.  ... 
doi:10.1007/978-3-0348-8268-2_9 fatcat:kwo7jxkbxfb4xanzt2bp4kwzhe

3-Wise Exactly 1-Intersecting Families of Sets

Zsolt Katona
2005 Graphs and Combinatorics  
Let g(l, t, n) be the maximal size of an l-wise exaclty t-intersecting family that is not trivially t-intersecting.  ...  Let f(l,t,n) be the maximal size of a family F ⊂ 2 [n] such that any l ≥ 2 sets of F have an exactly t ≥ 1-element intersection.  ...  A simple theorem of Erdős, Ko and Rado [6] says that the maximum of |F| is 2 n−1 , if every two members of a family F ⊆ 2 [n] have a non-empty intersection.  ... 
doi:10.1007/s00373-004-0592-x fatcat:ewhee6jco5dqtdch7uk7kaigym
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