Colouring of Maximal $F$-free Subsets

Bill Sands

Colouring of Maximal $F$-free Subsets

Bill Sands

Abstract

For each finite poset $F$ with $|F| > 1$, $χ_{ac}(F)$ 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 $χ_{ac}(F)$ for each poset $F$, give one new result, and summarize what is known for posets $F$ with at most four elements.

Colouring of Maximal $F$-free Subsets

Abstract

For each finite poset

with

denotes the smallest integer

(if it exists) such that the elements of every finite poset

with

can be coloured with at most

colours so that every maximal

-free subset of

with more than one element gets at least two colours. In this note we discuss the problem of determining

for each poset

, give one new result, and summarize what is known for posets

with at most four elements.

Paper Structure (3 theorems)

This paper contains 3 theorems.

Key Result

Theorem 1

Let $F$ be a bounded poset with no interior splitting elements. Then $\chi_{ac}(F)\le 10$.

Theorems & Definitions (3)

Theorem 1
Theorem 2
Theorem 3