On Chromatic no. of 3K1-free graphs and R(3, k)

Abstract

Here we prove that if G has independence no. 2 and clique size omega with omega less than or equal to 11, then (1) chromatic no. is less than or equal to (omega2+12omega-13)/8, if omega is odd, and (2) chromatic no. is less than or equal to (omega2+10omega)/8, if omega is even. We further conjecture that the results are true in general for all omega. We also conjecture that (A) if omega is odd and R(3, omega) is even, then R(3, omega) = (omega2+8omega-9)/4, (B) if omega and R(3, omega) are both odd, then (omega2+8omega-13)/4, (C) if omega and R(3, omega) are both even, then R(3, omega) = (omega2+6omega)/4 and (D) if omega is even and R(3, omega) is odd, then (omega2+6omega-4)/4. Again we verify the results for omega less than or equal to 9.