New bounds on the Ramsey number r(I-m, L-n)

We investigate the Ramsey number r(I-m, L-n) which is the smallest natural number k such that every oriented graph on k vertices contains either an independent set of size m or a transitive tournament on n vertices. Continuing research by Larson and Mitchell and earlier work by Bermond we establish two new upper bounds for r(I-m, L-3) which are paramount in proving r(I-4, L-3) = 15 < 23 = r(I-5, L-3) and r(I-m, L-3) = Theta(m(2)/logm), respectively. We furthermore elaborate on implications of the latter on upper bounds for r(I-m, L-n). (C) 2020 Elsevier B.V. All rights reserved.