Resumen de Characterizing the extremal families in Erdős-Ko-Rado theorems
Gilad Chase, Neta Dafni, Yuval Filmus, Nathan Lindzey
A family F ⊆ [n]kof k-sets of an n-element set [n] = {1, 2, · · · , n} is t-intersecting if|x ∩ y| ≥ t for all x, y ∈ F. In 1961, Erd˝os, Ko, and Rado showed for all t ≤ k < n/2 thatthe largest t-intersecting families have size no greater than n−tk−t, and moreover, that theextremal families are precisely the canonically t-intersecting families FT , i.e., the familiesobtained by taking all k-sets containing a given t-set T ⊆ [n]. This seminal result hassince been generalized to a variety of other combinatorial domains collectively known asErd˝os–Ko–Rado combinatorics. In this area, spectral techniques have been quite effectivefor bounding the sizes of t-intersecting families of various combinatorial objects, but generalmethods for characterizing the extremal t-intersecting families for all n, t remain elusive.We present new spectral techniques for characterizing the extremal t-intersecting familiesof various combinatorial domains for small t. We use these techniques to prove a coupleof Erd˝os–Ko–Rado conjectures on the characterization of extremal 2-intersecting families,namely, for 2-intersecting families of permutations Sn, 2-intersecting families of perfectmatchings of K2n, and so-called partially 2-intersecting families of perfect matchings of thecomplete k-uniform hypergraph Kkkn on kn vertices for k fixed and n sufficiently larg
