A Simple Construction of Tournaments with Finite and Uncountable Dichromatic Number

TL;DR

This paper addresses the problem of constructing tournaments with prescribed dichromatic numbers, including uncountable cases under CH. It introduces a simple checkerboard-based framework that ties color classes to acyclic subtournaments via $c$-sparse partitions, enabling a finite construction with $\chi(\vec{T}_k)=k$ for every $k$ and a lower bound $\chi(\vec{K}^{(m)}_n) \ge \dfrac{nm}{n+2m-2}$ for oriented complete balanced $n$-partite graphs. It further provides an elementary uncountable construction under CH yielding $\chi(\vec{T}_{\aleph_1})=\aleph_1$, extending the finite method to the infinite realm. Collectively, these results offer accessible, constructive insights into dichromatic colorings and clarify connections between combinatorial partitions and colorability in both finite and infinite settings.

Abstract

The dichromatic number $χ(\vec{G})$ of a digraph $\vec{G}$ is the minimum number of colors needed to color the vertices $V(\vec{G})$ in such a way that no monochromatic directed cycle is obtained. In this note, for any $k\in \mathbb{N}$, we give a simple construction of tournaments with dichromatic number exactly equal to $k$. The proofs are based on a combinatorial lemma on partitioning a checkerboard which may be of independent interest. We also generalize our finite construction to give an elementary construction of a complete digraph of cardinality equal to the cardinality of $\mathbb{R}$ and having an uncountable dichromatic number. Furthermore, we also construct an oriented balanced complete $n$-partite graph $\vec{K}^{(m)}_n$, such that the minimum number of colors needed to color its vertices such that there is no monochromatic directed triangle is greater than or equal to $nm/(n+2m-2)$.

Figures (1)

• Figure 1: Optimal c-sparse partition of a $7 \times 7$ checkerboard, where each of the four color classes represents a c-sparse set.

Theorems & Definitions (21)

• theorem 2.1
• lemma 2.1
• proof
• theorem 2.2
• proof
• lemma 2.2
• proof
• proof : Proof of Theorem \ref{['sparse partition']}
• lemma 2.3
• proof