Abstract
We investigate the Ramsey number r(Im,Ln) 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(Im,L3) which are paramount in proving r(I4,L3)=15<23=r(I5,L3) and r(Im,L3)=Θ(m2∕logm), respectively. We furthermore elaborate on implications of the latter on upper bounds for r(Im,Ln).
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