Skip to main content
Log in

Ordered and Canonical Ramsey Numbers of Stars

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bondy, J.A., Murty, U.S.R.: Graph theory with applications. London: Macmillan 1976

    MATH  Google Scholar 

  2. Choudum, S.A., Ponnusamy, B.: Ordered Ramsey numbers. communicated

  3. Erdös, P., Rado, R.: On a combinatorial theorem. J. London Math. Soc. 25, 249–255 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  4. Lefmann, H., Rödl, V., Thomas R.: Monochromatic vs multicolored paths. Graphsand Combinatorics 8, 323–332 (1992)

    Article  MATH  Google Scholar 

  5. Lefmann, H., Rödl, V.: On canonical Ramsey numbers for complete graphs vs paths. J. Combin. Theory Ser. B 58, 1–13 (1993)

    Article  Google Scholar 

  6. Lefmann, H., Rödl, V.: On Erdös Rado Numbers. Combinatorica 15, 85–104 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Moon, J.W.: Topics on tournaments. Holt Rinehart Winston (1968)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. A. Choudum.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03352992

Keywords

Navigation