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):
Preprint: maybe coming eventually
Code and checks:
Survey: S. P. Radziszowski, Small Ramsey Numbers, Electron. J. Combin. DS1 (rev. 18, 2026)

I have not published this or got it peer reviewed.

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