Embedding graphs into colored graphs

Hajnal, A.; Komjáth, P.

doi:10.1090/S0002-9947-1988-0936824-0

Embedding graphs into colored graphs
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by A. Hajnal and P. Komjáth

Trans. Amer. Math. Soc. 307 (1988), 395-409

DOI: https://doi.org/10.1090/S0002-9947-1988-0936824-0

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Corrigendum: Trans. Amer. Math. Soc. 332 (1992), 475.

Abstract:

If $X$ is a graph, $\kappa$ a cardinal, then there is a graph $Y$ such that if the vertex set of $Y$ is $\kappa$-colored, then there exists a monocolored induced copy of $X$; moreover, if $X$ does not contain a complete graph on $\alpha$ vertices, neither does $Y$. This may not be true, if we exclude noncomplete graphs as subgraphs. It is consistent that there exists a graph $X$ such that for every graph $Y$ there is a two-coloring of the edges of $Y$ such that there is no monocolored induced copy of $X$. Similarly, a triangle-free $X$ may exist such that every $Y$ must contain an infinite complete graph, assuming that coloring $Y$’s edges with countably many colors a monocolored copy of $X$ always exists.

References

J. Baumgartner and A. Hajnal, A proof (involving Martin’s axiom) of a partition relation, Fund. Math. 78 (1973), no. 3, 193–203. MR 319768, DOI 10.4064/fm-78-3-193-203
James E. Baumgartner, A new class of order types, Ann. Math. Logic 9 (1976), no. 3, 187–222. MR 416925, DOI 10.1016/0003-4843(76)90001-2
W. Wistar Comfort and Stylianos A. Negrepontis, Chain conditions in topology, Cambridge Tracts in Mathematics, vol. 79, Cambridge University Press, Cambridge-New York, 1982. MR 665100
Walter Deuber, Partitionstheoreme für Graphen, Comment. Math. Helv. 50 (1975), no. 3, 311–320. MR 401546, DOI 10.1007/BF02565753
Keith J. Devlin, Aspects of constructibility, Lecture Notes in Mathematics, Vol. 354, Springer-Verlag, Berlin-New York, 1973. MR 0376351
P. Erdős, F. Galvin, and A. Hajnal, On set-systems having large chromatic number and not containing prescribed subsystems, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vols. I, II, III, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp. 425–513. MR 0398876
P. Erdős and A. Hajnal, On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hungar. 17 (1966), 61–99. MR 193025, DOI 10.1007/BF02020444
P. Erdős and A. Hajnal, On decomposition of graphs, Acta Math. Acad. Sci. Hungar. 18 (1967), 359–377. MR 223261, DOI 10.1007/BF02280296
P. Erdős and A. Hajnal, On chromatic number of infinite graphs, Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic Press, New York, 1968, pp. 83–98. MR 0263693
P. Erdős, A. Hajnal, and L. Pósa, Strong embeddings of graphs into colored graphs, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vols. I, II, III, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp. 585–595. MR 0382049
P. Erdős, A. Hajnal, and B. Rothchild, “On chromatic number of graphs and set-systems” (Acta Math. Acad. Sci. Hungar. 17 (1966), 61–99) by Erdős and Hajnal, Cambridge Summer School in Mathematical Logic (Cambridge, 1971) Lecture Notes in Math., Vol. 337, Springer, Berlin, 1973, pp. 531–538. MR 0387103
Jon Folkman, Graphs with monochromatic complete subgraphs in every edge coloring, SIAM J. Appl. Math. 18 (1970), 19–24. MR 268080, DOI 10.1137/0118004
Ronald L. Graham, Rudiments of Ramsey theory, CBMS Regional Conference Series in Mathematics, vol. 45, American Mathematical Society, Providence, R.I., 1981. MR 608630
A. Hajnal and J. Pach, Monochromatic paths in infinite coloured graphs, Finite and infinite sets, Vol. I, II (Eger, 1981) Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 359–369. MR 818247
Péter Komjáth, A note on Hajnal-Máté graphs, Studia Sci. Math. Hungar. 15 (1980), no. 1-3, 275–276. MR 681447
Péter Komjáth, The colouring number, Proc. London Math. Soc. (3) 54 (1987), no. 1, 1–14. MR 872247, DOI 10.1112/plms/s3-54.1.1
Péter Komjáth and Vojtěch Rödl, Coloring of universal graphs, Graphs Combin. 2 (1986), no. 1, 55–60. MR 1117132, DOI 10.1007/BF01788077
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
J. Nešetřil and V. Rödl, A Ramsey graph without triangles exists for any graph without triangles, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vols. I, II, III, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp. 1127–1132. MR 0392695
Jaroslav Ne et il and Vojtěch Rödl, Partitions of vertices, Comment. Math. Univ. Carolinae 17 (1976), no. 1, 85–95. MR 412044
Jaroslav Ne et il and Vojtěch Rödl, On a probabilistic graph-theoretical method, Proc. Amer. Math. Soc. 72 (1978), no. 2, 417–421. MR 507350, DOI 10.1090/S0002-9939-1978-0507350-7
Saharon Shelah, Colouring without triangles and partition relation, Israel J. Math. 20 (1975), 1–12. MR 427073, DOI 10.1007/BF02756751

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