Colouring of Maximal $F$-free Subsets

Abstract

For each finite poset $F$ with $|F| > 1$, $\chi_{ac}(F)$ denotes the smallest integer $n$ (if it exists) such that the elements of every finite poset $P$ with $|P| > 1$ can be coloured with at most $n$ colours so that every maximal $F$-free subset of $P$ with more than one element gets at least two colours. In this note we discuss the problem of determining $\chi_{ac}(F)$ for each poset $F$, give one new result, and summarize what is known for posets $F$ with at most four elements.