Combinatorics Seminar-Injective Proofs of the Erdos-Ko-Rado and Hilton-Milner Theorems Speaker: GlennHurlbert, VCU

Abstract: Let F be a family of r-subsets of {1, 2, ..., n}. We say that F is intersecting if every pair of its sets intersect. The special case when some element (its center) is in each of its sets is called a star. The Erdos-Ko-Rado Theorem (1961 [really 1938]) states that, when n > 2r, the largest intersecting family is a star. The Hilton-Milner Theorem (1967) states that, when n > 2r, the largest non-star intersecting family is a near-star: a star with an extra set not containing its center. Vikram Kamat and I recently devised the first injective proofs of these classical results. I will share them with you in this talk. I'll also discuss some recent work with Fishel, Kamat, and Meagher on variations of the Erdos-Ko-Rado Theorem with other structures, such as permutations, trees, etc.