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SUMMARY:Karen Meagher\, Mahsa N. Shirazi and Sarobidy Razafimahatratra (Un
 iversity of Regina)
DTSTART:20210803T171500Z
DTEND:20210803T184500Z
DTSTAMP:20260416T224806Z
UID:CCM2021/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CCM2021/8/">
 The Ratio Bound for Erdős-Ko-Rado Type Theorems</a>\nby Karen Meagher\, M
 ahsa N. Shirazi and Sarobidy Razafimahatratra (University of Regina) as pa
 rt of Carleton Combinatorics Meeting 2021\n\n\nAbstract\nIn this talk Kare
 n Meagher will introduce the Ratio Bound\, also called Hoffman's Bound or 
 Delsarte's Bound. This is an algebraic bound the size of the maximum cocli
 que/independent set in a graph. This bound has been used to prove many var
 iations of the Erdős-Ko-Rado theorem. She will  outline some examples of 
 such problems and will show how it can be used effectively for such proble
 ms.\n\nThen Sarobidy Razafimahatratra will use the Hoffman Bound to prove 
 the Erdős-Ko-Rado\ntheorem for transitive groups.  A set of permutations 
 $\\mathcal{F}$ of a finite transitive group $G\\leq\n\\operatorname{Sym}(\
 \Omega)$ is <em>intersecting</em> if any two\npermutations in $\\mathcal{F
 }$ agree on an element of $\\Omega$. The\ntransitive group $G$ is said to 
 have the <em>Erdős-Ko-Rado (EKR)\nproperty</em> if any intersecting set o
 f $G$ has size at most\n$\\frac{|G|}{|\\Omega|}$. The alternating group $\
 \operatorname{Alt}(4)$ acting on the six\n$2$-subsets of $\\{1\,2\,3\,4\\}
 $ is an example of groups without the EKR\nproperty. This means that trans
 itive groups do not always have the EKR\nproperty.    Given a transitive g
 roup $G\\leq\\operatorname{Sym}(\\Omega)$\, we are interested in finding t
 he size and\nstructure of the largest intersecting sets in $G$. In this ta
 lk\, we will\nuse the Hoffman bound to prove the EKR property for some fam
 ilies of\ntransitive groups.\n\nFinally\, Mahsa Nasrollahi Shirazi will pr
 ove an extension of the Erdős-Ko-Rado theorem to set-wise intersecting pe
 rfect matchings. Two perfect matchings $P$ and $Q$ of a graph on $2k$ vert
 ices are said to be set-wise $t$-intersecting if there exist edges $P_{1}\
 , \\ldots\, P_{t}$ in $P$ and $Q_{1}\, \\ldots\, Q_{t}$ in $Q$ such that t
 he union of edges $P_{1}\, \\ldots\, P_{t}$ has the same set of vertices a
 s the union of $Q_{1}\, \\ldots\, Q_{t}$. We define a graph for which find
 ing a maximum coclique is equivalent to finding a largest family of set-wi
 se $t$-intersecting perfect matchings for $t=2\,3$. In this approach we us
 e the ratio bound to  find bounds on the size of a maximum coclique in thi
 s graph.\n
LOCATION:https://researchseminars.org/talk/CCM2021/8/
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