Upper and lower bounds for $F_v(4,4;5)$

Abstract

In this note we give a computer assisted proof showing that the unique $(5,3)$-Ramsey graph is the unique $K_5$-free graph of order 13 giving $F_v(3,4;5) \leq 13$, then we prove that $17 \leq F_v(2,2,2, 4; 5) \leq F_v(4, 4; 5) \leq 23$. This improves the previous best bounds $16 \leq F_v(4, 4; 5) \leq 25$ provided by Nenov and Kolev.