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Claim. J. A. Bondy and M. Simonovits, Cycles of even length in graphs, J. Combin. Theory Ser. B 16 (1974), no. 2, 97--105, DOI 10.1016/0095-8956(74)90052-5 (received 21 February 1973; issued April 1974, whose nominal first day is this page's date). Its Theorem 1 (p. 98) reads: "Theorem 1. If $e(G^n)>100k\,n^{1+1/k}$, then $C^{2l}\subset G^n$ for every integer $l\in[k,kn^{1/k}]$", where $G^n$ is a graph on $n$ vertices, $e(G^n)$ its number of edges and $C^{2l}$ the cycle of length $2l$. With $l=k$ it gives $\mathrm{ex}(n,C_{2k})\le100k\,n^{1+1/k}$ for every $k\ge2$. For Problem 1021 with $k=3$, the graph $G_3$ joins each of the three pairs of $\{y_1,y_2,y_3\}$ to its own vertex, so $G_3$ is the six-cycle $C_6$, and the theorem with $k=3$ gives $\mathrm{ex}(n,C_6)\le300n^{4/3}=300n^{3/2-1/6}$; so $c_3=1/6$ works. The claim value is proved: the result proves the statement for $k=3$.

Covers. The case $k=3$, where $G_3=C_6$, with $c_3=1/6$.

Depends on. Nothing in this wiki; the identification $G_3=C_6$ and the substitution $k=3$ are elementary and written above.

Acceptance. Refereed: published in the Journal of Combinatorial Theory, Series B, cited with its venue above. No reviewed evidence is listed: the site's PROVED label settles the whole problem and credits Conlon and Lee and Janzer, whose claim pages carry it, and the site's remark on $k=3$, which credits Erdős [Er64c] and Bondy and Simonovits, misprints the bound as $\mathrm{ex}(n,C_6)\ll n^{7/6}$. The site also credits Erdős's 1964 proceedings paper [Er64c] with this case; there Erdős stated the even-cycle bound without proof. Bondy and Simonovits (p. 97) describe Erdős's theorem as published without proof, and Erdős writes in his 1974 survey (equation 5) that he never published a proof and that Bondy and Simonovits have proved it, so that statement has no claim page of its own. Read depth: Theorem 1, the deduction of Theorem 1 from Theorem 1* and the quoted theorem of Erdős (pp. 97--98) are checked clause by clause; the proofs (Sections 2--3, pp. 99--104) are followed for structure only, and nothing is independently reviewed by this project.