On the Ramsey numbers \(r(S_n,K_6-3K_2)\)

Abstract

For every connected graph \( F \) with \( n \) vertices and every graph \( G \) with chromatic surplus \( s(G) (n-1)(\chi(G)-1) + s(G), \) where \( \chi(G) \) denotes the chromatic number of \( G \). If this lower bound is attained, then \( F \) is called \( G \)-good. For all connected graphs \( G \) with at most six vertices and \( \chi(G) > 4 \), every tree \( T_n \) of order \( n > 5 \) is \( G \)-good. In the case of \( \chi(G) = 3 \) and \( G \neq K_6 – 3K_2 \), every non-star tree \( T_n \) is \( G \)-good except for some small \( n \), whereas \( r(S_n, G) \) for the star \( S_n = K_{1,n-1} \) in a few cases differs by at most 2 from the lower bound. In this note we prove that the values of \( r(S_n, K_6 – 3K_2) \) are considerably larger for sufficiently large \( n \). Furthermore, exact values of \( r(S_n, K_6 – 3K_2) \) are obtained for small \( n \).