1) V1 For every real c>0, infinitely many n satisfy R(C4,K1,n)≤n+sqrt(n)-c.
open, filed Tue Aug 25 2026 06:34:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Six-role fleet passed: independent arrowing transcription agrees both ways; complete K4 has a red C4, the empty graph has none and has a 3-leaf blue star on four vertices; false-premise control preflights red/restatement; negation is the exact open infinite-occurrence statement; literature and final Jig dedupe agree. Whole attacks used the minimum-degree reformulation, C4 extremal edge bounds, projective-plane polarity graphs, prime-power exact cases, prime-gap interpolation, and the competing conjecture R=n+ceil(sqrt n)+{0,1}. No route proves or refutes infinitely many deficits, and no finite certificate was misfiled as asymptotic progress.
Scope. The highlighted second question only. Ramsey arrowing is expanded directly over every red graph on floor(n+sqrt(n)-c) vertices: either four distinct vertices form a red 4-cycle or one center has at least n blue neighbors.