Exact values and improved bounds on [formula omitted]-neighborly families of boxes

Abstract

A finite family F of d-dimensional convex polytopes is called k-neighborly if d−k⩽dim(C∩C′)⩽d−1 for any two distinct members C,C′∈F. In 1997, Alon initiated the study of the general function n(k,d), which is defined to be the maximum size of k-neighborly families of standard boxes in Rd. Based on a weighted count of vectors in {0,1}d, we improve a recent upper bound on n(k,d) by Alon, Grytczuk, Kisielewicz, and Przesławski for any positive integers d and k with d⩾k+2. In particular, when d is sufficiently large and k⩾0.123d, our upper bound on n(k,d) improves the bound ∑i=1k2i−1di+1 shown by Huang and Sudakov exponentially.Furthermore, we determine that n(2,4)=9, n(3,5)=18, n(3,6)=27, n(4,6)=37, n(5,7)=74, and n(6,8)=150. The stability result of Kleitman’s isodiametric inequality plays an important role in the proofs.