Monochromatic configurations on a circle

TL;DR

The paper studies two-colourings of the circle and seeks monochromatic k-point configurations with prescribed arc-lengths, formalizing Ramsey k-tuples and proving that the only Ramsey tuples are the $(k,2)$-powers with $d_i=\frac{2^{k-i}}{2^{k}-1}$. The core approach links Ramsey properties to Beatty sequence partition questions via balanced sequences and Fraenkel's conjecture, using discrete reductions to uniform colourings and modular polygon arguments. The authors also establish robust (nearly-Ramsey) variants, construct a counterexample to a majority-version, and provide extensive computational evidence up to $k\le 7$ (and higher-accuracy checks for specific steps up to $k=20$). These results tie combinatorial geometry on the circle to number-theoretic partition problems and raise several open questions regarding extensions, permutations, and density thresholds with potential implications for vector balancing and discrepancy theory.

Abstract

If we two-colour a circle, we can always find an inscribed triangle with angles $(\fracπ{7},\frac{2π}{7},\frac{4π}{7})$ whose three vertices have the same colour. In fact, Bialostocki and Nielsen showed that it is enough to consider the colours on the vertices of an inscribed heptagon. We prove that for every other triangle $T$ there is a two-colouring of the circle without any monochromatic copy of $T$. More generally, for $k\geq 3$, call a $k$-tuple $(d_1,d_2,\dots,d_k)$ with $d_1\geq d_2\geq \dots \geq d_k>0$ and $\sum_{i=1}^k d_i=1$ a Ramsey $k$-tuple if the following is true: in every two-colouring of the circle of unit perimeter, there is a monochromatic $k$-tuple of points in which the distances of cyclically consecutive points, measured along the arcs, are $d_1,d_2,\dots,d_k$ in some order. By a conjecture of Stromquist, if $d_i=\frac{2^{k-i}}{2^k-1}$, then $(d_1,\dots,d_k)$ is Ramsey. Our main result is a proof of the converse of this conjecture. That is, we show that if $(d_1,\dots,d_k)$ is Ramsey, then $d_i=\frac{2^{k-i}}{2^k-1}$. We do this by finding connections of the problem to certain questions from number theory about partitioning $\mathbb{N}$ into so-called Beatty sequences. We also disprove a majority version of Stromquist's conjecture, study a robust version, and discuss a discrete version.

Figures (4)

• Figure 1: A set of 5 blue points dividing the circle into arcs proportional to $1:2:4:8:16$.
• Figure 2: Arithmetic sequences for $\alpha_1=\frac{7}{4}$, $\alpha_2=\frac{7}{2}$, $\alpha_3=\frac{7}{1}$, and the corresponding balanced sequence with densities $\frac{4}{7}$, $\frac{2}{7}$, $\frac{1}{7}$.
• Figure 3: If we follow the red $k$-gon, we jump over each blue interval exactly once.
• Figure 4: A counterexample for the majority version. The black dots are the vertices of a regular 16-gon, depicted only to ease the comparison of distances in the figure.

Theorems & Definitions (19)

• Conjecture 1.1: Stromquist's conjecture
• Theorem 1.2
• Lemma 1.3
• Conjecture 1.4: Fraenkel's conjecture
• Theorem 1.5
• Theorem 1.6
• Conjecture 2.2
• Lemma 2.3
• proof : Proof of Theorem \ref{['thm:fraenkelspec']}
• proof : Proof of Lemma \ref{['lemma:uniform']}