On monochromatic pairs of nondecreasing diameters

Adam O'Neal¹ ; Michael W. Schroeder¹

¹Marshall University

The electronic journal of combinatorics, Tome 26 (2019) no. 2

Résumé

Let $n$, $m$, $r$, $t$ be positive integers and $\Delta:[n]\to[r]$. We say $\Delta$ is $(m,r,t)$-permissible if there exist $t$ disjoint $m$-sets $B_1,\dots,B_t$ contained in $[n]$ for which $|\Delta(B_i)|=1$ for each $i=1,2,\dots,t$, $\max(B_i) < \min(B_{i+1})$ for each $i=1,2,\dots,t-1$, and $\max(B_i)-\min(B_i) \leq \max(B_{i+1})-\max(B_{i+1})$ for each $i=1,2,\dots,t-1$. Let $f(m,r,t)$ be the smallest such $n$ so that all colorings $\Delta$ are $(m,r,t)$-permissible. In this paper, we show that $f(2,2,t)=5t-4$.

Détail

DOI : 10.37236/8003

Classification : 05D10, 11B75, 11B50
Mots-clés : colorings