# Exact value for Ramsey number of \(k\)-cycle and star

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For two graphs \( G\) and \( H\), let \( r(G,H)\) denote the smallest integer \( m\) satisfying the property that if the edges of the complete graph \( K_m\) are colored in red and blue, then there is either a subgraph isomorphic to \( G\) with all red edges or a subgraph isomorphic to \( H\) with all blue edges. The classical Ramsey numbers are those for the complete graphs and are denoted by \( r(s,t)= r(K_s, K_t)\).

Together with Burr {1], Erdös, Faudree, Rousseau, and Schelp proposed the following problem:

** Problem**

Determine \(r(C_4,K_{1,n})\).
It is known that

\(\displaystyle n + \lceil \sqrt n \rceil +1 \geq r(C_4,K_{1,n}) \geq n + \sqrt{n}- 6 n^{11/40}, \)

where the upper bound can be easily derived from the Turán number of \( C_4\) and the lower bound can be found in [1].
Füredi shows (unpublished) that
\( r(C_4, K_{1,n})= n + \lceil \sqrt n \rceil \) holds infinitely often.