Exact Ramsey Number $R(5,5)$

The Ramsey number $R(5,5)$ is the least $N$ such that every red–blue coloring of the edges of the complete graph $K_N$ contains a monochromatic $K_5$. Its existence follows from Ramsey's theorem, and computing it is a canonical finite special case of Ramsey's general problem rather than a separately dated conjecture; the determination of such small Ramsey numbers has been a benchmark challenge in combinatorics for decades [Radziszowski2026Ramsey].

Current knowledge places the value in the interval $43\le R(5,5)\le 46$. The lower bound reflects an explicit red–blue coloring of $K_{42}$ with no monochromatic $K_5$, while the upper bound $R(5,5)\le 46$ is due to Angeltveit and McKay [AngeltveitMcKay2024R55]. Radziszowski's survey of small Ramsey numbers tracks these and related values [Radziszowski2026Ramsey].

Four candidate values remain. Settling the problem requires either exhibiting a coloring on more than $42$ vertices that avoids a monochromatic $K_5$, or certifying that no such coloring exists; until one of these is achieved, the exact value of $R(5,5)$ remains open.