One of the more recent generalizations of the Erdős-Ko-Rado theorem, formulated by Holroyd, Spencer and Talbot, defines the Erdős-Ko-Rado property for graphs in the following manner: for a graph G, vertex v ∈ G and some integer r 1 denote the family, called a star. Then G is said to be r-EKR if no intersecting subfamily of J (r) (G) is larger than the largest star in J (r) (G). In this paper, we prove that if G is a disjoint union of chordal graphs, including at least one singleton, then G is r-EKR if r μ(G) 2 , where μ(G) is the minimum size of a maximal independent set. We also prove Erdős-Ko-Rado results for chains of complete graphs, which are special chordal graphs obtained by blowing up edges of a path into complete graphs, as well as prove preliminary results for trees.
We consider the following generalization of the seminal Erdős-Ko-Rado theorem, due to Frankl [5]. For some k ≥ 2, let F be a k-wise intersecting family of r-subsets of an n element set X, i.e. for any F1, . . . , F k ∈ F,We prove a stability version of this theorem, analogous to similar results of Dinur-Friedgut, Keevash-Mubayi and others for the Erdős-Ko-Rado theorem. The technique we use is a generalization of Katona's circle method, initially employed by Keevash, which uses expansion properties of a particular Cayley graph of the symmetric group.
We consider the following higher-order analog of the Erdős-Ko-Rado theorem. For positive integers r and n with r ≤ n, let M r n be the family of all matchings of size r in the complete graph K2n. For any edge e ∈ E(K2n), the family M r n (e), which consists of all sets in M r n containing e is called the star centered at e. We prove that if r < n and A ⊆ M r n is an intersecting family of matchings, then |A| ≤ |M r n (e)|, where e ∈ E(K2n). We also prove that equality holds if and only if A is a star. The main technique we use to prove the theorem is an analog of Katona's elegant cycle method.
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