On Monochromatic Pairs with Nondecreasing Diameters

Authors

Adam O'Neal
Michael W. Schroeder, Marshall UniversityFollow

Document Type

Article

Publication Date

5-3-2019

Abstract

Let n m r t be positive integers and Δ : [n] → [r]. We say Δ is (m, r, t) - permissible if there exist t disjoint m-sets B₁,…,B_t contained in [n] for which

1. |Δ(B_i)| = 1 for each i = 1,2,…, t.
2. max(B_i) < min(B_i+1) for each i = 1,2,…, t − 1, and
3. max(B_i) − min(B_i) ≤ max(B_i+1) − max(B_i+1) for each i = 1, 2,…, t − 1.

Let f(m ,r, t) be the smallest such n so that all colorings Δ are (m, r, t)-permissible. In this paper, we show that f(2, 2, t) = 5t − 4.

Comments

Copyright © The authors. Released under the CC BY-ND license (International 4.0).

Recommended Citation

O'Neal, Adam; Schroeder, Michael W. On monochromatic pairs with nondecreasing diameters. Electron. J. Combin. 26 (2019), no. 2, Paper 2.15, 12 pp.