Stop one edge short
Clone a vertex at the threshold
Start from a target-free coloring on R−1 vertices, clone one vertex, and leave the edge between the pair uncolored. The partial coloring on K_R−e is still target-free.
Paper project · Ramsey theory
Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors
Overview
After existence becomes unavoidable, count what remains forced.
A Ramsey number marks the first size at which every edge-coloring contains a prescribed monochromatic clique. Threshold Ramsey multiplicity asks a second question at that exact size: among all colorings, how few monochromatic target cliques can there be?
The paper develops upper bounds for multicolor, off-diagonal threshold multiplicities. The small R(3,3)=6 example below makes the distinction concrete: every red-blue coloring of K6 contains a monochromatic triangle, and the best coloring still contains two.
The upper-bound construction delays the first target until one final edge is colored. By cloning a vertex and leaving the edge to its clone missing, the coloring remains target-free on K_R−e. Once e is colored, every newly forced clique contains that single backbone edge, so the count reduces to a smaller clique count among common neighbors in that same color.
Color the last edge
Force every new target through e
Whichever color the missing edge receives, any target appearing for the first time must contain that edge; otherwise it would already have existed in K_R−e.
Count in the matching-color neighborhood
Reduce K_s to K_{s−2}
A monochromatic K_s through a backbone edge of color i is exactly that edge plus a color-i K_{s−2} among the common color-i neighbors of its endpoints, where extremal clique counting supplies the bound.
Threshold multiplicity and the last edge
Count what survives—and where every copy is anchored.
The displayed K6 coloring is one optimal example, with one red and one blue triangle. The interactive below shows the cloning construction itself: begin with a triangle-free red–blue K5, clone one vertex, and color the single missing edge. Either color creates exactly two triangles of that color, both through the final edge.
R(3,3) = 6
Exactly two monochromatic triangles
This displayed coloring attains m₂(3)=2. The paper studies upper bounds in broader multicolor and off-diagonal settings.
Citation
Use or reference this work.
The title and citation reflect the revised 2026 version of the preprint.
BibTeX citation
@article{christopherson2025threshold,
title = {Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors},
author = {Christopherson, Bryce and Steinhaus, Casia},
year = {2025},
eprint = {2501.18869},
archivePrefix = {arXiv},
primaryClass = {math.CO},
url = {https://arxiv.org/abs/2501.18869}
}