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Claim. Theorem 3 (PDF p. 2; the article carries no printed folios) of Yali Wu, Yongqi Sun, Rui Zhang and Stanisław P. Radziszowski, Ramsey numbers of $C_4$ versus wheels and stars, Graphs Combin. 31 (2015), no. 6, 2437--2446: for every prime power $q\ge3$,

$$R(C_4,K_{1,q^2-2})=q^2+q-1,$$

and for even $q$, $R(C_4,K_{1,q^2-k-1})=q^2+q-k$ for $0\le k\le q$, $k\notin\{1,q-1\}$. These are values of the function $f(n)=R(C_4,S_n)$ of Problem 552. The lower bounds come from graphs built on the simple polarity graph of Abreu, Balbuena and Labbate, the upper bounds from a counting bound on $C_4$-free graphs. The paper's wheel results (its Theorems 2 and 4) concern $R(C_4,W_m)$ and are not part of this claim. The statement is recorded on the library home wu_2015_ramsey_numbers_c_4_versus_wheels_stars.

Covers. The value of $f(n)$ at $n=q^2-2$ for every prime power $q\ge3$, and at $n=q^2-k-1$ for every even prime power $q$ and $0\le k\le q$, $k\notin\{1,q-1\}$. The value at every other $n$, and the second question, whether $f(n)\le n+\sqrt n-c$ for infinitely many $n$, are not settled by it.

Depends on. Nothing in this wiki; the theorem rests on the paper's own counting bound and constructions.

Acceptance. Refereed: the paper is a journal publication in Graphs and Combinatorics, volume 31, number 6 (2015), published online 24 January 2015, the date this page is named by, the refereed evidence. The site's curator refers to this paper in the commentary on exact values, but the site's label OPEN settles neither the problem nor a declared part of it, so reviewed is not listed.