Cliques in block graphs of designs and orthogonal arrays

Abstract

The Erdos-Ko-Rado [EKR] Theorem for intersecting families is a fundamental result in combinatorics, particularly in extremal set theory. This theorem not only establishes an upper bound on the size of the largest intersecting family but also characterizes the families that attain this bound—–these are known as maximal canonically intersecting. Recent work by Balogh, Das, Delcourt, Liu, and Sharifzadeh delves into intersecting families across permutations, hypergraphs, and vector spaces, revealing that nearly all such families within these structures are a subset of a maximal canonically intersecting family [1]. Building on these insights, this thesis extends the examination to block graphs of designs and orthogonal arrays. Through a comprehensive analysis of intersecting families within designs, we introduce a ratio between canonically intersecting families and non-canonical ones, demonstrating that almost all intersecting families in designs are canonical. This method, adaptable to orthogonal arrays OA(m, n) for sufficiently large n, is complemented by a conjecture proposing a second proof inspired by the methods given by Balogh et al. Notably, these results exclude symmetric and affine designs.