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Claim. Xuemei Zhang, Yaojun Chen and T.C. Edwin Cheng, Some values of Ramsey numbers for $C_4$ versus stars, Finite Fields Appl. 45 (2017), 73--85, Theorem 6 (p. 75): "Let $q\ge4$ be an even prime power and $t=1,0,-2$. Then $R(C_4,K_{1,(q-1)^2+t})=(q-1)^2+q+t$." Theorem 7 (p. 75): "Let $q\ge5$ be an odd prime power, $t=2,4,\ldots,2\lceil\frac q4\rceil$. Then $R(C_4,K_{1,q(q-1)-t})=q^2-t$." These are values of the function $f(n)=R(C_4,S_n)$ of Problem 552. The lower bounds come from a $C_4$-free graph on $q^2-1$ vertices over $GF(q)$ with a few vertices deleted. The paper places every value on the lines $n+\lceil\sqrt n\rceil$ and $n+\lceil\sqrt n\rceil+1$, and records $f(40)=47$ and $f(38)=45$ ($q=7$ in Theorem 7) as the values new to its table of $2\le n\le50$. For $q=5$ the proof of Theorem 7 reads two values off results it cites (Parsons's $f(16)=21$ and a computer determination for $n=18$). The statements are recorded on the result pages Theorem 6 and Theorem 7 of the library home zhang_2017_some_values_ramsey_numbers_c_4_versus_stars.

Covers. The value of $f(n)$ at $n=(q-1)^2+t$, $t=1,0,-2$, for every even prime power $q\ge4$, and at $n=q(q-1)-t$, $t=2,4,\ldots,2\lceil q/4\rceil$, for every odd prime power $q\ge5$. 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; the paper asks (p. 76) whether every value is $n+\lceil\sqrt n\rceil$ or one more, which would answer the second question in the negative.

Depends on. Parsons 1975, whose Theorem 2 the proof of Theorem 7 uses at $q=5$ together with a cited computer value; otherwise the paper's own constructions and lemmas.

Acceptance. Refereed: the paper is a journal publication in Finite Fields and Their Applications, volume 45 (May 2017), the refereed evidence; the issue carries no day, so this page is dated to the first day of that month. 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. The statements are checked against the publisher's text; the proofs are read for structure only.