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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hypergraphs and Ramseyian theorems


Author: V. Chvátal
Journal: Proc. Amer. Math. Soc. 27 (1971), 434-440
MSC: Primary 05.55
MathSciNet review: 0270972
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Abstract: A $ k$-graph is an ordered couple $ (V,E)$ where $ V$ is a set and $ E$ a set of $ k$-tuples of elements of $ V$; thus, a $ 2$-graph is an ordinary graph. If the notions of the independent set and the chromatic number are generalized for $ k$-graphs then one can ask what is the least number of edges in a $ k$-graph having $ p$ vertices and no independent set of size $ b$ (the problem of Turán) and what is the least number of edges in a $ k$-graph whose chromatic number exceeds a given number (the generalized problem of Erdös and Hajnal). As in graphs, there is a relationship between independent sets and chromatic numbers--actually, our results for the first problem are applicable to the second one. The theorems of Ramsey's type are, in fact, theorems on the chromatic number of certain $ k$-graphs; thus, the results for the problem of Erdös and Hajnal yield lower bounds for the general ``Ramseyian numbers".


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0270972-9
PII: S 0002-9939(1971)0270972-9
Keywords: Hypergraph, Turán's problem, covering problems, independent sets, chromatic number, Ramsey type theorems
Article copyright: © Copyright 1971 American Mathematical Society