A set of vertices $Xsubseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $xin X$ is either isolated in the induced subgraph $langle Xrangle$ or else has a private neighbor $yin Vsetminus X$ that is adjacent to $x$ and to no other vertex of $X$. The emph{irredundant Ramsey number} $s(m,n)$ is the smallest $N$ such that in every red-blue coloring of the edges of the complete graph of order $N$, either the blue subgraph contains an $m$-element irredundant set or the red subgraph contains an $n$-element irredundant set. The emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ yields an $m$-element irredundant set in the blue subgraph or an $n$-element independent set in the red subgraph. In this paper, we first improve the upper bound of $t(3,n)$; using this result, we confirm that a conjecture proposed by Chen, Hattingh, and Rousseau, that is, $lim_{nrightarrow infty}frac{t(m,n)}{r(m,n)}=0$ for each fixed $mgeq 3$, is true for $mleq 4$. At last, we prove that $s(3,9)$ and $t(3,9)$ are both equal to $26$.
Download