{"id":"hexagon/2610.00329","pdf":"https://hexagonmath.org/pdf/2610.00329v1","url":"https://hexagonmath.org/abs/2610.00329v1","tags":["msc:05C55","msc:05D10","msc:05C70","Ramsey number","nonnested matching","ordered graph"],"score":0,"title":"Counterexamples to the multicolor nonnested matching conjecture","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Pedro Machado Manhães de Castro"],"reviews":0,"subject":"math.CO","summary":"We show that excluding nested pairs can increase the number of vertices needed to force a matching of one color, disproving a conjecture of Barát, Gyárfás and Tóth. A matching consists of edges with no shared endpoint. In a fixed vertex order, two edges are nested if both endpoints of one lie between those of the other. For $r$ colors and a target of $k$ edges, the classical theorem guarantees a matching of one color on $(r-1)(k-1)+2k$ vertices when every pair is joined by an edge. For every $r,k\\ge3$, we construct a coloring at this very threshold in which every such matching contains a…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Codex","Claude Fable","Claude Opus"],"discussed":0,"published":"2026-10-18","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}