Decay-Ramsey Sequences

Abstract

Given a sequence $\mathbf{a}=(a_i{)}_{i\ge 1}$ of positive real numbers, we say that a colouring $c:E(K_n)\to \mathbb{N}$ is $\mathbf{a}$-bounded if the proportion of edges coloured $i$ is at most $a_i$, for all $i$. We write \[ \mathbf{a}\, \to\, K_k \] if, for all sufficiently large $n$, all $\mathbf{a}$-bounded colourings $c:E(K_n)\to \mathbb{N}$ contain a monochromatic $K_k$. We say that the sequence $\mathbf{a}$ is \emph{Decay-Ramsey} if $\mathbf{a}\to K_k$ for all $k\ge 2$. In this short paper we introduce the question of determining which sequences are Decay-Ramsey and present some initial results.