Alon's transmitting problem and multicolor Beck--Spencer Lemma

TL;DR

This paper extends a combinatorial result about the Hamming graph by generalizing Alon's problem to multicolor cases and employing a multicolor floating variable method to prove the existence of a vertex with specific distance properties.

Contribution

It introduces a multicolor extension of the Beck--Spencer Lemma and applies it to prove a generalized Alon-type problem for Hamming graphs with more than two colors.

Findings

Generalization of Alon's problem to q-ary Hamming graphs

Extension of the Beck--Spencer Lemma using multicolor floating variable method

Existence of a vertex with prescribed distance properties in q-ary Hamming graphs

Abstract

The Hamming graph $H(n,q)$ is defined on the vertex set $\{1,2,\ldots,q\}^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon (1992) proved that for any sequence $v_1,\ldots,v_b$ of $b=\lceil\frac n2\rceil$ vertices of $H(n,2)$, there is a vertex whose distance from $v_i$ is at least $b-i+1$ for all $1\leq i\leq b$. In this note, we prove that for any $q\geq 3$ and any sequence $v_1,\ldots,v_b$ of $b=\lfloor(1-\frac1q)n\rfloor$ vertices of $H(n,q)$, there is a vertex whose distance from $v_i$ is at least $b-i+1$ for all $1\leq i\leq b$. Alon used a lemma due to Beck and Spencer (1983) which, in turn, was based on the floating variable method introduced by Beck and Fiala (1981) who studied combinatorial discrepancies. For our proof, we extend the Beck--Spencer Lemma by using a multicolor version of the floating variable method due to Doerr and Srivastav (2003).