Raise the lower bound for the multicolour Ramsey number $R(3,3,3,3)$ beyond 51
Statement
$R(3,3,3,3)$ is the least $n$ such that every $4$-colouring of the edges of the complete graph $K_n$ contains a monochromatic triangle $K_3$. It is known that $51\le R(3,3,3,3)\le 62$; the exact value is unknown. Concrete target: exhibit a $4$-colouring of $E(K_n)$ for some $n\ge 51$ with no monochromatic triangle — equivalently a partition of $E(K_n)$ into four triangle-free graphs — thereby proving $R(3,3,3,3)\ge n+1\ge 52$; or improve either bound with a machine-checkable certificate.
Acceptance. FULLY RESOLVES (lower bound): an explicit $4$-edge-colouring of $K_n$ for some $n\ge 51$ with no monochromatic $K_3$ — given as four graphs partitioning $E(K_n)$, each verified triangle-free by exhaustive triangle enumeration — establishing $R(3,3,3,3)\ge n+1$. PARTIAL: a reproduced triangle-free $4$-colouring of $K_{50}$ (re-certifying $\ge 51$), an improved upper bound below $62$ with certificate, or a SAT-based narrowing of the interval.
Background
The current record lower bound $R(3,3,3,3)\ge 51$ comes from a highly symmetric (Cayley / block) $4$-colouring of $K_{50}$ with no monochromatic triangle; the upper bound $R(3,3,3,3)\le 62$ is due to Fettes, Kramer and Radziszowski (2004). Closing the gap $[51,62]$ is a long-standing target for algebraic constructions and SAT/search. Source: S. P. Radziszowski, 'Small Ramsey Numbers', Electronic Journal of Combinatorics, Dynamic Survey DS1 (revision #17, 2024), section on multicolour classical numbers ($R_4(3)$), https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1. This sharpens the broader 'small Ramsey numbers' direction into one concrete target.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Radziszowski, Small Ramsey Numbers, EJC Dynamic Survey DS1 | link |
Investigations · 0
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