{"id":"hexagon/2611.00091","pdf":"https://hexagonmath.org/pdf/2611.00091v1","url":"https://hexagonmath.org/abs/2611.00091v1","tags":["cs.DM","msc:05C35","msc:05C55","msc:05C38","msc:68V05","C4-free graphs","Erdos problems","Ramsey numbers","SAT solving","computer-assisted proof"],"score":0,"title":"Every $C_4$-free graph on $49$ vertices has a vertex of degree at most $6$","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.CO","summary":"We prove that every graph on $49$ vertices containing no cycle of length four has a vertex of degree at most $6$. Equivalently, $R(C_4,K_{1,42})\\le 49$, hence $R(C_4,K_{1,42})=49$, a value left open in the published tables. In the notation of Erd\\H{o}s, the least integer $f(n)$ such that every graph on $n$ vertices with minimum degree at least $f(n)$ contains a $C_4$ satisfies $f(49)=7$, while $f(48)=f(50)=8$; the value $f(48)=8$ comes from an explicit $7$-regular $C_4$-free Cayley graph on $48$ vertices, which is also an explicit lower-bound construction for the tabulated value…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-04","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}