Pure Pairs. II. Excluding All Subdivisions of A Graph

Abstract

We prove for every graph H there exists ɛ > 0 such that, for every graph G with |G|≥2, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least ɛ|G| neighbours, or there are two disjoint sets A, BV(G) with |A|,|B|≥ɛ|G| such that no edge joins A and B. It follows that for every graph H, there exists c>0 such that for every graph G, if no induced subgraph of G or its complement is a subdivision of H, then G has a clique or stable set of cardinality at least |G|c. This is related to the Erdős-Hajnal conjecture.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon, J. Pach, R. Pinchasi, R. Radoičić and M. Sharir: Crossing patterns of semi-algebraic sets, J. Combinatorial Theory, Ser. A 111 (2005), 310–326.

    MathSciNet  Article  Google Scholar 

  2. [2]

    M. Bonamy, N. Bousquet and S. Thomassé: The Erdős-Hajnal conjecture for long holes and antiholes, SIAM J. Discrete Math 30 (2016), 1159–1164.

    MathSciNet  Article  Google Scholar 

  3. [3]

    N. Bousquet, A. Lagoutte and S. Thomassé: The Erdős-Hajnal conjecture for paths and antipaths, J. Combinatorial Theory, Ser. B 113 (2015), 261–264.

    MathSciNet  Article  Google Scholar 

  4. [4]

    K. Choromanski, D. Falik, A. Liebenau, V. Patel and M. Pilipczuk: Excluding hooks and their complements, Electronic Journal of Combinatorics 25 #P3.27, 2018.

  5. [5]

    M. Chudnovsky: The Erdős-Hajnal conjecture a survey, J. Graph Theory 75 (2014), 178–190.

    MathSciNet  Article  Google Scholar 

  6. [6]

    M. Chudnovsky, A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. III. Long holes, Combinatorica 37 (2017), 1057–72.

    MathSciNet  Article  Google Scholar 

  7. [7]

    M. Chudnovsky, A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. XI. Orientations, European Journal of Combinatorics 76 (2019), 53–61, arXiv:1711.07679.

    MathSciNet  Article  Google Scholar 

  8. [8]

    M. Chudnovsky, A. Scott, P. Seymour and S. Spirkl: Pure pairs. I. Trees and linear anticomplete pairs, Advances in Math 375 (2020), 107396.

    MathSciNet  Article  Google Scholar 

  9. [9]

    M. Chudnovsky and P. Seymour: Excluding paths and antipaths, Combinatorica 35 (2015), 389–412.

    MathSciNet  Article  Google Scholar 

  10. [10]

    M. Chudnovsky and Y. Zwols: Large cliques or stable sets in graphs with no fouredge path and no five-edge path in the complement, J. Graph Theory 70 (2012), 449–472.

    MathSciNet  Article  Google Scholar 

  11. [11]

    P. Erdős: Some remarks on the theory of graphs, Bull. Amer. Math. Soc 53 (1947), 292–294.

    MathSciNet  Article  Google Scholar 

  12. [12]

    P. Erdős and A. Hajnal: On spanned subgraphs of graphs, Graphentheorie und Ihre Anwendungen (Oberhof, 1977), https://www.renyi.hu/~p_erdos/1977-19.pdf

    Google Scholar 

  13. [13]

    P. Erdős and A. Hajnal: Ramsey-type theorems, Discrete Applied Mathematics 25 (1989), 37–52.

    MathSciNet  Article  Google Scholar 

  14. [14]

    P. Erdős and G. Szekeres: A combinatorial problem in geometry, Compositio Mathematica 2 (1935), 463–470.

    MathSciNet  MATH  Google Scholar 

  15. [15]

    J. Fox: A bipartite analogue of Dilworth's theorem, Order 23 (2006), 197–209.

    MathSciNet  Article  Google Scholar 

  16. [16]

    J. Fox and J. Pach: Erdős-Hajnal-type results on intersection patterns of geometric objects, in: Horizon of Combinatorics (G. O. H. Katona et al., eds.), Bolyai Society Studies in Mathematics, Springer, 79–103, 2008.

    Chapter  Google Scholar 

  17. [17]

    A. Gyérfés: On Ramsey covering-numbers, Coll. Math. Soc. Jénos Bolyai, in: Infinite and Finite Sets, North Holland/American Elsevier, New York (1975), 10.

    Google Scholar 

  18. [18]

    A. Liebenau, M. Pilipczuk, P. Seymour and S. Spirkl: Caterpillars in ErdősHajnal, J. Combinatorial Theory, Ser. B 136 (2019), 33–43, arXiv:1810.00811.

    MathSciNet  Article  Google Scholar 

  19. [19]

    A. Pawlik, J. Kozik, T. Krawczyk, M. Lasoń, P. Micek, W. T. Trotter and B. Walczak: Triangle-free intersection graphs of line segments with large chromatic number, J. Combinatorial Theory, Ser. B 105 (2014), 6–10.

    MathSciNet  Article  Google Scholar 

  20. [20]

    V. Rödl: On universality of graphs with uniformly distributed edges, Discrete Math 59 (1986), 125–134.

    MathSciNet  Article  Google Scholar 

  21. [21]

    A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. I. Odd holes, J. Combinatorial Theory, Ser. B 121 (2016), 68–84.

    MathSciNet  Article  Google Scholar 

  22. [22]

    A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes, J. Combinatorial Theory, Ser. B 132 (2018), 180–235, arXiv:1509.06563.

    MathSciNet  Article  Google Scholar 

  23. [23]

    A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. IX. Rainbow paths, Electronic J. Combinatorics 24 #P2.53, 2017.

  24. [24]

    A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. X. Holes with specific residue, Combinatorica 39 (2019), 1105–1132

    MathSciNet  Article  Google Scholar 

  25. [25]

    D. P. Sumner: Subtrees of a graph and chromatic number, in: The Theory and Applications of Graphs, (G. Chartrand, ed.), John Wiley & Sons, New York (1981), 557–576.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alex Scott.

Additional information

Supported by NSF grant DMS-1550991. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under grant number W911NF-16-1-0404.

Supported by a Leverhulme Trust Research Fellowship.

Supported by ONR grant N00014-14-1-0084, AFOSR grant A9550-19-1-0187, and NSF grants DMS-1265563 and DMS-1800053.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chudnovsky, M., Scott, A., Seymour, P. et al. Pure Pairs. II. Excluding All Subdivisions of A Graph. Combinatorica 41, 379–405 (2021). https://doi.org/10.1007/s00493-020-4024-1

Download citation

Mathematics Subject Classification (2010)

  • 05C75
  • 05C69