Intersection theorems for t-valued functions

AbstractThis paper investigates the maximum possible size of families ℱ of t-valued functions on an n-element set S = {1, 2, . . . , n}, assuming any two functions of ℱ agree in sufficiently many places. More precisely, given a family ℬ of k-element subsets of S, it is assumed for each pair h, g ∈ ℱ that there exists a B in ℬ such that h = g on B. If ℬ is ‘not too large’ it is shown that the maximal families have tn−k members

Some of my Favourite Problems in Number Theory, Combinatorics, and Geometry

To the memor!l of m!l old friend Professor George Sved.I heard of his untimel!l death while writing this paper

Compression and Erdos-Ko-Rado graphs

For a graph G and integer r >= 1 we denote the collection of independent r-setsof G by I^(r)(G). If v is in V(G) then I^(r)_v(G) is the collection of all independent r-sets containing v. A graph G is said to be r-EKR, for r >= 1, iff no intersecting family A of I^(r)(G) is larger than max_{v in V(G)} |I^(r)_v(G)|. There are various graphs that are known to have this property; the empty graph of order n >= 2r (this is the celebrated Erdos-Ko-Rado theorem), any disjoint union of atleast r copies of K_t for t >= 2, and any cycle. In this paper we show how these results can be extended to other classes of graphs via a compression proof technique. In particular we extend a theorem of Berge, showing that any disjoint union of at least r complete graphs, each of order at least two, is r-EKR. We also show that paths are r-EKR for all r >= 1

Towards a general theory of Erdős-Ko-Rado combinatorics

2014 Summer.Includes bibliographical references.In 1961, Erdős, Ko, and Rado proved that for a universe of size n ≥ 2k a family of k-subsets whose members pairwise intersect cannot be larger than n-1/k-1. This fundamental result of extremal combinatorics is now known as the EKR theorem for intersecting set families. Since then, there has been a proliferation of similar EKR theorems in extremal combinatorics that characterize families of more sophisticated objects that are largest with respect to a given intersection property. This line of research has given rise to many interesting combinatorial and algebraic techniques, the latter being the focus of this thesis. Algebraic methods for EKR results are attractive since they could potentially give rise to a unified theory of EKR combinatorics, but the state-of-the-art has been shown only to apply to sets, vector spaces, and permutation families. These categories lie on opposite ends of the stability spectrum since the stabilizers of sets and vector spaces are large as possible whereas the stabilizer of a permutation is small as possible. In this thesis, we investigate a category that lies somewhere in between, namely, the perfect matchings of the complete graph. In particular, we show that an algebraic method of Godsil's can be lifted to the more general algebraic framework of Gelfand pairs, giving the first algebraic proof of the EKR theorem for intersecting families of perfect matchings as a consequence. There is strong evidence to suggest that this framework can be used to approach the open problem of characterizing the maximum t-intersecting families of perfect matchings, whose combinatorial proof remains illusive. We conclude with obstacles and open directions for extending this framework to encompass a broader spectrum of categories