Hypergraphs and Ramseyian theorems
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- by V. Chvátal
- Proc. Amer. Math. Soc. 27 (1971), 434-440
- DOI: https://doi.org/10.1090/S0002-9939-1971-0270972-9
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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".References
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Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 434-440
- MSC: Primary 05.55
- DOI: https://doi.org/10.1090/S0002-9939-1971-0270972-9
- MathSciNet review: 0270972