More Constructive Lower Bounds on Classical Ramsey Numbers

Abstract

We present several new constructive lower bounds for classical Ramsey numbers. In particular, the inequality $R(k,s+1) \geq R(k,s)+2k-2$ is proved for $k \geq 5$. The general construction permits us to prove that, for all integers k, l, with $k \geq 5$ and $l \geq 3$, the connectivity of any Ramsey-critical $(k,l)$-graph is at least k, and if $k \geq l-1 \geq 1$, $k \geq 3$ and $(k,l) \neq (3,2)$, then such graphs are Hamiltonian. New concrete lower bounds for Ramsey numbers are obtained, some with the help of computer algorithms, including: $R(5,17) \geq 388$, $R(5,19) \geq 411$, $R(5,20) \geq 424$, $R(6,8) \geq 132$, $R(6,12) \geq 263$, $R(7,8) \geq 217$, $R(7,9) \geq 241$, $R(7,12) \geq 417$, $R(8,17) \geq 961$, $R(9,10) \geq 581$, $R(12,12) \geq 1639$, and also one three-color case $R(8,8,8) \geq 6079$.