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• W2969237952 abstract "A family $mathcal{A}$ of sets is said to be intersecting if every two sets in $mathcal{A}$ intersect. An intersecting family is said to be emph{trivial} it its sets have a common element. A graph $G$ is said to be $r$-EKR if at least one of the largest intersecting families of independent $r$-element sets of $G$ is trivial. Let $alpha(G)$ and $omega(G)$ denote the independence number and the clique number of $G$, respectively. Hilton and Spencer recently showed that if $G$ is the vertex-disjoint union of a cycle ${_*C}$ raised to the power $k^*$ and $s$ cycles ${_1C}, dots, {_sC}$ raised to the powers $k_1, dots, k_s$, respectively, $1 leq r leq alpha(G)$, and $$minbig(omega(_1C^{k_1}), dots, omega(_sC^{k_s})big) geq 2k^* + 1,$$ then $G$ is $r$-EKR. They had shown that the same holds if ${_*C}$ is replaced by a path and the condition on the clique numbers is relaxed to $$minbig(omega(_1C^{k_1}), dots, omega(_sC^{k_s})big) geq k^* + 1.$$ We use the classical Shadow Intersection Theorem of Katona to obtain a short proof of each result for the case where the inequality for the minimum clique number is strict." @default.
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• W2969237952 date "2019-08-23" @default.
• W2969237952 modified "2023-09-27" @default.
• W2969237952 title "On the Hilton-Spencer intersection theorems for unions of cycles" @default.
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