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, B ⊆ V(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.
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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.
M. Bonamy, N. Bousquet and S. Thomassé: The Erdős-Hajnal conjecture for long holes and antiholes, SIAM J. Discrete Math 30 (2016), 1159–1164.
N. Bousquet, A. Lagoutte and S. Thomassé: The Erdős-Hajnal conjecture for paths and antipaths, J. Combinatorial Theory, Ser. B 113 (2015), 261–264.
K. Choromanski, D. Falik, A. Liebenau, V. Patel and M. Pilipczuk: Excluding hooks and their complements, Electronic Journal of Combinatorics 25 #P3.27, 2018.
M. Chudnovsky: The Erdős-Hajnal conjecture a survey, J. Graph Theory 75 (2014), 178–190.
M. Chudnovsky, A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. III. Long holes, Combinatorica 37 (2017), 1057–72.
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.
M. Chudnovsky, A. Scott, P. Seymour and S. Spirkl: Pure pairs. I. Trees and linear anticomplete pairs, Advances in Math 375 (2020), 107396.
M. Chudnovsky and P. Seymour: Excluding paths and antipaths, Combinatorica 35 (2015), 389–412.
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.
P. Erdős: Some remarks on the theory of graphs, Bull. Amer. Math. Soc 53 (1947), 292–294.
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
P. Erdős and A. Hajnal: Ramsey-type theorems, Discrete Applied Mathematics 25 (1989), 37–52.
P. Erdős and G. Szekeres: A combinatorial problem in geometry, Compositio Mathematica 2 (1935), 463–470.
J. Fox: A bipartite analogue of Dilworth's theorem, Order 23 (2006), 197–209.
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.
A. Gyérfés: On Ramsey covering-numbers, Coll. Math. Soc. Jénos Bolyai, in: Infinite and Finite Sets, North Holland/American Elsevier, New York (1975), 10.
A. Liebenau, M. Pilipczuk, P. Seymour and S. Spirkl: Caterpillars in ErdősHajnal, J. Combinatorial Theory, Ser. B 136 (2019), 33–43, arXiv:1810.00811.
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.
V. Rödl: On universality of graphs with uniformly distributed edges, Discrete Math 59 (1986), 125–134.
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.
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.
A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. IX. Rainbow paths, Electronic J. Combinatorics 24 #P2.53, 2017.
A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. X. Holes with specific residue, Combinatorica 39 (2019), 1105–1132
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.
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.
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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
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