The Ramsey number of the fan F_4 is 18

Let F_n = K_1 + nK_2 be the fan (friendship graph) consisting of n triangles sharing one common vertex. We show that R(F_4, F_4) = 18. Previously 17 ≤ R(F_4, F_4) ≤ 24 was known, and the dynamic survey of Radziszowski lists only the upper bound 28. The lower bound is an explicit 2-colouring of K_17. For the upper bound we bound every colour degree of an (F_4, F_4)-free colouring by 11, split according to the maximum colour degree D ∈ {9, 10, 11} of a hypothetical colouring of K_18, and refute each case by a SAT solver. All unsatisfiability claims are certified by LRAT proofs that were checked with the formally verified checker cake_lpr. As a by-product we independently confirm R(F_3, F_4) = 17, recently obtained by Jiang and Shao. Files: fanF4.pdf is the note (6 pages). fanF4_data.zip contains the colourings, the four final CNF formulas with SHA-256 checksums, the scripts that generate and check them, and the verification logs. The LRAT proofs (1.3 GB in total) are not included; they can be regenerated deterministically with CaDiCaL 2.1.3 from the supplied CNF files and checked with cake_lpr. The search code, the computations and a first draft of the note were produced with the help of an AI assistant (Claude, Anthropic). The author is responsible for the content.