Abstract
A canonical version of Ramsey’s Theorem proved by Erdös and Rado, implies that given any acyclic digraph D, there exists a least integer ρ c (D) = n, such that every arc colouring (with an arbitrary number of colours) of the transitive tournament TT n contains a canonically coloured D (in the sense of Erdös-Rado). It follows that if P m is a directed path and D is an acyclic digraph, then there exists a least integer ρ*(P m , D) = n such that every arc coloring of TT n , with an arbitrary number of colours, contains either a P m with no two arcs of the same colour or a monochromatic D. Recently, Lefmann, Rödl and Thomas [4], Lefmann and Rödl [5] have studied the numbers ρ*(P n , P m ) and ρ*(P n , TT m ). In this paper we find ρ*(P n , S m ), where S m is the out-star and give bounds for ρ c (S m, n ) where S m, n is the directed star with m in-arcs and n out-arcs at the centre.
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References
Bondy, J.A., Murty, U.S.R.: Graph theory with applications. London: Macmillan 1976
Choudum, S.A., Ponnusamy, B.: Ordered Ramsey numbers. communicated
Erdös, P., Rado, R.: On a combinatorial theorem. J. London Math. Soc. 25, 249–255 (1950)
Lefmann, H., Rödl, V., Thomas R.: Monochromatic vs multicolored paths. Graphsand Combinatorics 8, 323–332 (1992)
Lefmann, H., Rödl, V.: On canonical Ramsey numbers for complete graphs vs paths. J. Combin. Theory Ser. B 58, 1–13 (1993)
Lefmann, H., Rödl, V.: On Erdös Rado Numbers. Combinatorica 15, 85–104 (1995)
Moon, J.W.: Topics on tournaments. Holt Rinehart Winston (1968)
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Choudum, S.A., Ponnusamy, B. Ordered and Canonical Ramsey Numbers of Stars. Graphs and Combinatorics 13, 147–158 (1997). https://doi.org/10.1007/BF03352992
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DOI: https://doi.org/10.1007/BF03352992