Jessica De Silva, California State University, Stanislaus
Authors: Cashous Bortner, Paola Campos, Jessica De Silva, Jeffrey Venable
2023 AWM Research Symposium
Pure and Applied Talks by Mathematicians Enhancing Diversity in Graduate Education (EDGE) [Organized by Quiyana M. Murphy and Sofía Martínez Alberga]

The Erdős-Ko-Rado Theorem is a pivotal result in extremal set theory which gives an upper bound on the number of sets of a fixed size that are pairwise intersecting. Of particular interest is the straightforward construction of an intersecting family that attains this bound by collecting all sets of the specified size which contain some fixed element. In 2005, Holroyd, Spencer, and Talbot formulated an EKR property for graphs related to intersecting families of independent sets. This property has a corresponding construction, called an r-star, which takes all independent sets of size r containing a fixed vertex of the graph. A graph is called r-EKR if the maximum size of an intersecting family of size r independent sets is equal to the size of the largest r-star in the graph. A longstanding conjecture of Holroyd and Talbot suggests that all graphs are r-EKR under certain conditions on r. Talbot suggests that if a counterexample to this conjecture exists, it may lie in the class of well-covered graphs. In this talk, we will present results related to maximum-sized r stars in very well-covered graphs.

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