Chromatic Number of Complete Continuous Graphs

Prove or refute the conjecture that the one-half-chromatic number of the continuous complete graph Γ(Kₙ) equals ⌈n/2⌉ for every natural number n.

Background

The paper defines the one-half-chromatic number using covers by radius-one-half balls whose intersecting balls receive different colors. It constructs a coloring of Γ(Kₙ) using ⌈n/2⌉ colors, proving the corresponding upper bound, but explicitly states that it cannot rigorously prove the matching lower bound.

References

We conjecture that the $\frac{1}{2}$-chromatic number of $\Gamma(K_n)$ is $\lceil\frac{n}{2}\rceil$. Though we cannot provide a rigorous proof for the lower bound of this conjecture, the following proposition establishes the upper bound.

— Open Problems in Continuous Graphs  (2501.14554 - Grigoriev et al., 24 Jan 2025) in Section 3, subsection “Chromatic number,” paragraph “Colouring a complete continuous graph”