Using computational techniques we derive six new upper bounds on the
classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <=
68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are
improvements by one over the previously best known bounds.
Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on
n vertices without independent sets of order k. The new upper bounds on R(3,k)
are obtained by completing the computation of the exact values of e(3,k,n) for
all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower
bounds on e(3,k,n) for most of the open cases for 10 <= k <= 15. The
enumeration of all graphs witnessing the values of e(3,k,n) is completed for
all cases with k <= 9. We prove that the known critical graph for R(3,9) on 35
vertices is unique up to isomorphism. For the case of R(3,10), first we
establish that R(3,10) = 43 if and only if e(3,10,42) = 189, or equivalently,
that if R(3,10) = 43 then every critical graph is regular of degree 9. Then,
using computations, we disprove the existence of the latter, and thus show that
R(3,10) <= 42.Comment: 28 pages (includes a lot of tables); added improved lower bound for
R(3,11); added some note
