LOWER BOUNDS FOR SOME RAMSEY NUMBERS (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2004)
For the Ramsey number $R(p_1,\ldots,p_r)$, $r\geq2$, we prove that \[ R(p_1,\ldots,p_r) > \bigl( R(p_1,\ldots,p_s)-1\bigr) \bigl( R(p_{s+1},\ldots,p_r)-1\bigr), \] $s\in\{1,\ldots,r-1\}$. This inequality generalizes a result obtained by Robertson (Theorem 1) and improves the lower bounds for some Ramsey numbers.