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Claim. The site's wording asks for $R_3(C_n)\le4n-3$ for every cycle length $n\ge3$. At $n=3$ the cycle is the triangle, $C_3=K_3$, and $R_3(C_3)$ is the three-color Ramsey number $R(3,3,3)$. Greenwood and Gleason determined

$$R(3,3,3)=17,$$

so $R_3(C_3)=17>9=4\cdot3-3$ and the universal statement fails at its first instance. The disproof needs only the strict inequality $R(3,3,3)>9$, which is elementary: joining two copies of the two-colored $K_5$ without a monochromatic triangle (the pentagon in one color, the pentagram in the other) by all crossing edges in the third color gives a $3$-coloring of $K_{10}$ with no monochromatic triangle, so $R(3,3,3)\ge11$; the problem page records that check. The exact value is the refereed result recorded here as the claimant's. The corrected Statement, for $n>3$, is not touched by this page; its state is recorded on the problem page and on the partial claim pages Kohayakawa, Simonovits and Skokan 2005 and Benevides and Skokan 2008.

Why it is rejected. It answers the site's wording, not the corrected statement.

Depends on. Nothing in this wiki; the result is the paper's own theorem.