On subsets of the hypercube with prescribed Hamming distances

Abstract

A theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in {0, 1} n with diameter d has cardinality at most that of a Hamming ball of radius d/2. In this paper, we give an algebraic proof of Kleitman's Theorem, by carefully choosing a pseudo-adjacency matrix for certain Hamming graphs, and applying the Cvetković bound on independence numbers. This method also allows us to prove several extensions and generalizations of Kleitman's Theorem to other allowed distance sets, in particular blocks of consecutive integers that do not necessarily grow linearly with n. We also improve on a theorem of Alon about subsets of F n p whose difference set does not intersect {0, 1} n nontrivially.2 n−1 0 + · · · + n−1 t , for d = 2t + 1.. Both inequalities are sharp. When d = 2t, the upper bound is attained by a Hamming ball of radius d/2, for instance, a collection of binary vectors having at most t 1-coordinates. For d = 2t + 1, an optimal example is given by the Cartesian product of {0, 1} and the (n − 1)dimensional Hamming ball of radius t, or alternatively, a Hamming ball of radius d/2 centered at (1/2, 0, · · · , 0). Kleitman's Theorem can be reduced to the celebrated Katona intersection theorem [18], using the squashing operation (see [2] or [14]). Generalizations of this theorem have been studied for the n-dimensional grid [m] n with Hamming distance [3,13]; as well as [m] n and the n-dimensional torus Z n m with Manhattan distance [1,5,8].

Intersective sets over abelian groups

“…Furthermore, we carefully compute the spectrum of certain matrices and obtain some improved upper bounds. In particular, when J = {0, 1} and G = F p , we show a new upper bound D Fp (J, N) ≤ ( 1 2 + o( 1))(p − 1) N , which improves on a theorem of Huang, Klurman and Pohoata [12]. The main tools of this part are basic representation theory and quantitative versions of the central limit theorem.…”

“…M is called a pseudo-adjacency matrix of X if M is a real symmetric n × n matrix such that M ij = 0 whenever ij ∈ E(X). The following theorem is due to Cvetković (also known as the inertia bound); see for example [12,Corollary 2.5].…”

“…Huang, Klurman, and Pohoata [12,Theorem 1.3] first converted the problem to finding the independence number of a (weighted) Cayley graph, and then computed the eigenvalues of the pseudo-adjacency matrix and applied Cvetković bound to obtain an upper bound on the independence number. They also remarked at the end of [12,Section 4] that they believed a more careful analysis would improve the constant 1−1/2e to 1/2. We confirm their remark in Corollary 1.7 and some of our main results are inspired by their work.…”

“…(2) Let G = Z n and let k be a divisor of n such that 5,6,7,8,9,12,13,14,15,16,17,20,22,24,26,28,31,32,33,34,35,36,39,40,41,42,43,46,47…”