Ramsey Numbers and a New Bound for R(3,3,6)

Ramsey Numbers and a New Bound for R(3,3,6) Colour every line between n points red, blue or green. How many points force a red triangle, a blue triangle or a green clique of six? That number is the Ramsey number R(3,3,6). It was known to lie between 61 and 91; in this video we prove, with nothing but careful counting, that it is at most 88. Chapters: 0:00 Six people at a party 0:52 Five can avoid it, six cannot 2:36 Colourings and cliques 3:23 Ramsey numbers in general 4:23 A third colour 5:34 Somewhere between 61 and 91 5:58 One point's view: R(3,3,6) ≤ 92 7:25 Counting lines: R(3,3,6) ≤ 91 8:43 The road to 88 9:03 Paint it purple 9:50 Purple triangles are rare 11:17 Counting cherries two ways 11:50 Two points joined in green 12:55 One number per point 14:03 The parabola: R(3,3,6) ≤ 88 14:59 Looking back Paper (PDF): https://lachy-dauth.github.io/Ramsey_... Preprint: maybe coming eventually Code and checks: https://github.com/Lachy-Dauth/ramsey... Survey: S. P. Radziszowski, Small Ramsey Numbers, Electron. J. Combin. DS1 (rev. 18, 2026) I have not published this or got it peer reviewed.