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One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three

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Polcyn J.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 2

443-464

Let P denote a 3-uniform hypergraph consisting of 7 vertices a, b, c, d, e, f, g and 3 edges {a, b, c}, {c, d, e}, and {e, f, g}. It is known that the r-color Ramsey number for P is R(P; r) = r + 6 for r ≤ 9. The proof of this result relies on a careful analysis of the Turán numbers for P. In this paper, we refine this analysis further and compute the fifth order Turán number for P, for all n. Using this number for n = 16, we confirm the formula R(P; 10) = 16.

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Short paths in 3-uniform quasi-random hypergraphs

100%

Polcyn J.

Discussiones Mathematicae Graph Theory

2004

tom 24

nr 3

469-484

Frankl and Rödl [3] proved a strong regularity lemma for 3-uniform hypergraphs, based on the concept of δ-regularity with respect to an underlying 3-partite graph. In applications of that lemma it is often important to be able to "glue" together separate pieces of the desired subhypergraph. With this goal in mind, in this paper it is proved that every pair of typical edges of the underlying graph can be connected by a hyperpath of length at most seven. The typicality of edges is defined in terms of graph and hypergraph neighborhoods, and it is shown that all but a small fraction of edges are indeed typical.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Polcyn J.

1 2017

1 2004

Wyniki wyszukiwania

One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three

Short paths in 3-uniform quasi-random hypergraphs