Ramsey Numbers for Partially-Ordered Sets

Abstract

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.

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References

  1. 1.

    Alon, N., Frankl, P., Lovász, L.: The chromatic number of Kneser hypergraphs. Trans. Am. Math. Soc. 298(1), 359–370 (1986)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Axenovich, M., Walzer, S.: Boolean lattices: Ramsey properties and embeddings. Order 34(2), 287–298 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Balko, M., Cibulka, J., Král, K, Kynčl, J.: Ramsey numbers of ordered graphs. Electron Notes Discrete Math. 49, 419–424 (2015)

    Article  Google Scholar 

  4. 4.

    Beineke, L.W., Schwenk, A.J.: On a bipartite form of the Ramsey problem. In: Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), pp. 17–22 (1975)

  5. 5.

    Choudum, S., Ponnusamy, B.: Ordered Ramsey numbers. Discret. Math. 247 (1-3), 79–92 (2002)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cibulka, J., Gao, P., Krčál, M., Valla, T., Valtr, P.: On the geometric Ramsey number of outerplanar graphs. Discrete Comput. Geom. 53(1), 64–79 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Conlon, D.: A new upper bound for the bipartite Ramsey problem. J. Graph Theory 58(4), 351–356 (2008)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Conlon, D., Fox, J., Lee, C., Sudakov, B.: Ordered Ramsey numbers. J. Comb. Theory, Series B 122, 353–383 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cox, C., Stolee, D.: Ordered Ramsey numbers of loose paths and matchings. Discret. Math. 339(2), 499–505 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    De Bonis, A., Katona, G.O.: Largest families without an r-fork. Order 24(3), 181–191 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    De Bonis, A., Katona, G.O., Swanepoel, K.J.: Largest family without abcd. J. Comb. Theory, Series A 111(2), 331–336 (2005)

    MathSciNet  Article  Google Scholar 

  12. 12.

    De Moura, L., Bjørner, N.: Z3: an efficient smt solver. In: Tools and Algorithms for the Construction and Analysis of Systems, pp. 337–340. Springer (2008)

  13. 13.

    Duffus, D., Kierstead, H.A., Trotter, W.T.: Fibres and ordered set coloring. J. Comb. Theory, Series A 58(1), 158–164 (1991)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Fox, J., Pach, J., Sudakov, B., Suk, A.: Erdős-szekeres-type theorems for monotone paths and convex bodies. Proc. Lond. Math. Soc. 105(5), 953–982 (2012)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Füredi, Z.: An upper bound on Zarankiewicz’ problem. Comb. Probab. Comput. 5(01), 29–33 (1996)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Goddard, W., Henning, M.A., Oellermann, O.R.: Bipartite Ramsey numbers and Zarankiewicz numbers. Discret. Math. 219(1), 85–95 (2000)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Griggs, J., Li, W.-T., Lu, L.: Diamond-free families. J. Comb. Theory (Ser. A) 119, 310–322 (2012)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Griggs, J.R., Li, W.-T.: The partition method for poset-free families. J. Comb. Optim. 25(4), 587–596 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Griggs, J.R., Li, W.-T.: Poset-free families and lubell-boundedness. J. Comb. Theory, Series A 134, 166–187 (2015)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Griggs, J.R., Lu, L.: On families of subsets with a forbidden subposet. Comb. Probab. Comput. 18(05), 731–748 (2009)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Grósz, D., Methuku, A., Tompkins, C.: An improvement of the general bound on the largest family of subsets avoiding a subposet. Order 34(1), 113–125 (2017)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Gunderson, D.S., Rödl, V., Sidorenko, A.: Extremal problems for sets forming boolean algebras and complete partite hypergraphs. J. Comb. Theory, Series A 88(2), 342–367 (1999)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Hattingh, J., Henning, M.: Bipartite Ramsey theory. Utilitas Mathematica 53, 217–230 (1998)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Irving, R.W.: A bipartite Ramsey problem and the Zarankiewicz numbers. Glasg. Math. J. 19(01), 13–26 (1978)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Johnston, T., Lu, L., Milans, K.G.: Boolean algebras and lubell functions. J. Comb. Theory Series A, July. to appear

  27. 27.

    Kierstead, H.A., Trotter, W.T.: A ramsey theoretic problem for finite ordered sets. Discret. Math. 63(2), 217–223 (1987)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Kramer, L., Martin, R., Young, M.: On diamond-free subposets of the Boolean lattice. J. Comb. Theory Ser. A 120(3), 545–560 (2013)

    MathSciNet  Article  Google Scholar 

  29. 29.

    McColm, G.L.: A ramseyian theorem on products of trees. J. Comb. Theory, Series A 57(1), 68–75 (1991)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Milans, K., Stolee, D., West, D.: Ordered Ramsey theory and track representations of graphs. J. Comb. 6(4), 445–456 (2015)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Moshkovitz, G., Shapira, A.: Ramsey theory, integer partitions and a new proof of the ERDős-Szekeres theorem. Adv. Math. 262, 1107–1129 (2014)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Nešetřil, J., Rödl, V.: Combinatorial partitions of finite posets and lattices—Ramsey lattices. Algebra Universalis 19(1), 106–119 (1984)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27, 544–548 (1928)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Stein, W. et al.: Sage: open source mathematical software. 7 December 2009 (2008)

  35. 35.

    Trotter, W.: Ramsey theory and partially ordered sets. In: Graham, R.L. et al. (eds.) Contemporary Trends in Discrete Mathmatics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 49, pp. 337–347 (1999)

  36. 36.

    West, D.: Introduction to Graph Theory. Prentice Hall, Inc., Upper Saddle River (1996)

    MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Mikhail Lavrov for recommending SAT solvers as a method for computing small Boolean Ramsey numbers. Thanks also to Maria Axenovich for discussing previous work on induced Boolean Ramsey numbers.

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Correspondence to Christopher Cox.

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The authors completed this work while at Iowa State University.

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Cox, C., Stolee, D. Ramsey Numbers for Partially-Ordered Sets. Order 35, 557–579 (2018). https://doi.org/10.1007/s11083-017-9449-9

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Keywords

  • Ramsey theory
  • Partially-ordered sets
  • Ordered Ramsey numbers
  • Ordered graphs