Set systems related to a house allocation problem

Abstract

We are given a set A of buyers, a set B of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping τ from A to B, and τ is strictly better than another house allocation τ ′ = τ if for every buyer i, τ ′ (i) does not come before τ (i) in the preference list of i. A house allocation is Pareto optimal if there is no strictly better house allocation.Let s(τ ) be the image of τ (i.e., the set of houses sold in the house allocation τ ). We are interested in the largest possible cardinality f (m) of the family of sets s(τ ) for Pareto optimal mappings τ taken over all sets of preference lists of m buyers. We improve the earlier upper bound on f (m) given by Asinowski, Keszegh and Miltzow [2], by making a connection between this problem and some problems in extremal set theory.

Combinatorics in the exterior algebra and the Bollobás Two Families Theorem

“…As an application of our results, we prove a new extension of the Two Families Theorem of Bollobás. We also affirmatively resolve a recent conjecture of Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patkós, Tompkins and Xiao [20].…”

“…In Section 5 below we use Theorem 2.12 to settle a conjecture of Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patkós, Tompkins, and Xiao [20].…”

“…In a recent preprint [20], Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patkós, Tompkins, and Xiao consider bounding the size of fully cross-intersecting pairs of families of sets, with fixed profile (a, b), under the additional assumption that one of the two families is also t-intersecting. For the t = 1 case, they deduce an upper bound of 1 where AK(n, a, t) denotes the maximum size of an a-uniform t-intersecting family A ⊆ [n], as determined by Ahlswede and Khachatrian [1].…”

“…In a recent preprint [20], Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patkós, Tompkins and Xiao consider bounding the size of fully cross-intersecting pairs of families of sets, with fixed profile (a, b), under the additional assumption that one of the two families is also t-intersecting.…”

“…In Section 5 we show that the size of pairs of families satisfying both the Two Family hypotheses and an intersection condition on the first family is bounded by the Ahlswede-Khachatrian bound on the size of t-intersecting families, as conjectured by Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patkós, Tompkins, and Xiao [20]. Finally, in Section 6 we collect some limiting examples and propose a few questions.…”