New values of small Ramsey numbers for fans and complete bipartite graphs

Let F_n = K_1 + nK_2 be the fan consisting of n triangles sharing one vertex. We show that R(F_4,F_4) = 18, R(F_3,F_5) = 21 and R(F_4,F_5) = 22. Previously 18 ≤ R(F_4,F_4) ≤ 24, 21 ≤ R(F_3,F_5) ≤ 26 and 22 ≤ R(F_4,F_5) ≤ 28 were known; the lower bounds come from constructions of Li and Rousseau and of Zhang, Broersma and Chen, and our contribution is the matching upper bounds. We also show R(K_{3,3},K_{3,4}) = 19 and R(K_{2,4},K_{3,5}) = 19, which are listed as 19–20 in Radziszowski's dynamic survey; here too the lower bounds were known. The upper bounds are computer assisted. For fans, a formula of Lin and Li for R(F_t,sK_2) bounds the colour degrees, and a case split on a maximum colour degree fixes the neighbourhood of one vertex. Each case, and each bipartite instance, is refuted by a SAT solver, and the unsatisfiability of every resulting formula is certified by an LRAT proof accepted by cake_lpr, a proof checker formally verified in CakeML. The encodings were checked semantically on small orders, the final formulas were audited by separate programs, and the same code reproduces fourteen known values and confirms the value R(F_3,F_4) = 17 recently reported by Jiang and Shao. This paper combines and supersedes the two earlier notes doi:10.5281/zenodo.23902750 (R(F_4,F_4) = 18) and doi:10.5281/zenodo.23902938 (the other four values), whose data archives contain all formulas, generators and cake_lpr logs. Files here: fans_bip_ejc.pdf (the paper, 10 pages) and supplement.zip (recomputation of known values, second independent checks, CNF audits and SHA-256 checksums). Not peer reviewed. The author used an AI assistant (Claude, Anthropic) to write all programs, including the checking programs, to run the computations, and to prepare a first draft of this paper. The author takes full responsibility for the content.