Ramsey theory on Steiner triples

Granath, E. and Gyárfás, András and Hardee, J. and Watson, T. and Wu, X. (2018) Ramsey theory on Steiner triples. Journal of Combinatorial Designs, 26 (1). pp. 5-11. ISSN 1063-8539

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Abstract

We call a partial Steiner triple system C (configuration) t-Ramsey if for large enough n (in terms of (Formula presented.)), in every t-coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic copy of C. We prove that configuration C is t-Ramsey for every t in three cases: C is acyclic every block of C has a point of degree one C has a triangle with blocks 123, 345, 561 with some further blocks attached at points 1 and 4 This implies that we can decide for all but one configurations with at most four blocks whether they are t-Ramsey. The one in doubt is the sail with blocks 123, 345, 561, 147. © 2017 Wiley Periodicals, Inc.

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