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Claim. $\mathrm{ex}(n;C_6)\gg n^{4/3}$ and $\mathrm{ex}(n;C_{10})\gg n^{6/5}$, the statement of Problem 572 for $k=3$ and $k=5$. Theorem 1 of the paper on the library's source card (p. 1091): the point-line incidence graph $G_8$ of a non-degenerate quadric in $P(4,q)$ is a regular bipartite graph of degree $q+1$ and girth $8$ on $1+q+q^2+q^3$ points and as many lines (p. 1092). Theorem 2: the graph $G_{12}$ on the points of a quadric in $P(6,q)$ and its distinguished lines is a regular bipartite graph of degree $q+1$ and girth $12$ on $(q+1)(1+q^2+q^4)$ points and as many lines (p. 1093). The paper states no extremal number; the step to the bounds is the elementary count recorded on the problem page. $G_8$ has $n=2(1+q+q^2+q^3)$ vertices, $\frac n2(q+1)$ edges and no cycle shorter than $8$, and $\frac n2\le(q+1)^3$, so $\mathrm{ex}(n;C_6)\ge2^{-4/3}n^{4/3}$ at these orders; $G_{12}$ has $n=2(q+1)(1+q^2+q^4)$ vertices, $\frac n2(q+1)$ edges and no cycle shorter than $12$, and $\frac n2\le(q+1)^5$, so $\mathrm{ex}(n;C_{10})\ge2^{-6/5}n^{6/5}$. Since $\mathrm{ex}(n;C_{2k})$ is nondecreasing in $n$ and consecutive prime powers differ by a factor at most $2$, both bounds hold for every $n$ with smaller constants. Bondy and Simonovits record the conjecture as known for $k=2,3,5$ in Remark 1 of [BoSi74] (p. 98), citing Benson among others; the site's commentary credits Benson with the cases $k=3$ and $k=5$.

Covers. The instances $k=3$ (the cycle $C_6$) and $k=5$ (the cycle $C_{10}$) of the statement for every $k\ge3$. Nothing for $k=4$ or any $k\ge6$, which remain open; the problem has no full claim.

Depends on. Nothing in this wiki: the constructions are the paper's, and the counting step is elementary and recorded on the problem page.

Acceptance. Refereed: Clark T. Benson, Minimal regular graphs of girths eight and twelve, Canad. J. Math. 18 (1966), 1091--1094, doi:10.4153/CJM-1966-109-8, a refereed journal; the publisher's record gives the year only, and this page's date is the first day of that year by the corpus's convention. No reviewed evidence is listed: the site labels the problem OPEN, and commentary on an open problem is not an acceptance.

Read depth. Theorems 1 and 2 and the counts on pp. 1092--1093 were read clause by clause; the proofs were not read, and nothing is independently reviewed in this corpus.