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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the chromatic number of subgraphs of a given graph
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by V. Rödl PDF
Proc. Amer. Math. Soc. 64 (1977), 370-371 Request permission


It is shown that for arbitrary positive integers m,n there exists a $\phi (m,n)$ such that if $\chi (G) \geqslant \phi (m,n)$, then G contains either a complete subgraph of size m or a subgraph H with $\chi (H) = n$ containing no ${C_3}$. This gives an answer to a problem of Erdös and Hajnal.
  • P. Erdős, Problems and results in chromatic graph theory, Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) Academic Press, New York, 1969, pp. 27–35. MR 0252273
  • P. Erdős, Some unsolved problems in graph theory and combinatorial analysis, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) Academic Press, London, 1971, pp. 97–109. MR 0277392
  • A. Hajnal, R. Rado and V. T. Sós (Editors), Infinite and finite sets, (Proc. Colloq., Készthely, 1973, dedicated to P. Erdös), North-Holland, Amsterdam, 1975. MR 50 #12526.
  • Oystein Ore, Theory of graphs, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society, Providence, R.I., 1962. MR 0150753
  • Peter Ungar and Blanche Descartes, Advanced Problems and Solutions: Solutions: 4526, Amer. Math. Monthly 61 (1954), no. 5, 352–353. MR 1528740, DOI 10.2307/2307489
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 64 (1977), 370-371
  • MSC: Primary 05C15
  • DOI:
  • MathSciNet review: 0469806