The Ramsey number R(5,5)
Find the smallest number of vertices such that every red/blue coloring of a complete graph’s edges contains a monochromatic five-vertex clique.
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Understanding the problem
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Find the smallest number of vertices such that every red/blue coloring of a complete graph’s edges contains a monochromatic five-vertex clique.
R(5,5) asks for the first complete graph size at which every red/blue edge coloring contains a monochromatic clique on five vertices. A coloring on 42 vertices that avoids both colors’ five-cliques shows R(5,5) is at least 43.
For an upper bound, the argument must exclude every avoiding coloring at the claimed size. Symmetry reductions, exhaustive searches, and computer assistance can help, but their completeness must be justified. Ordinary computation is not automatically an AI contribution.
Why it matters
This focused Ramsey problem is a test case for difficult bounds and exhaustive verification.
What counts as progress
A better justified bound or a fully checked exact value. A single coloring supplies only a lower bound.
What to watch for
- A better lower bound needs a larger explicitly checkable avoiding graph.
- A better upper bound needs a proof excluding all cases, with computational certificates where applicable.
- An exact value requires matching lower and upper bounds. A proposed coloring or AI-written proof is not sufficient on its own.
Background review: 2026-10-11. Selected sources:
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