An improved upper bound for Ramsey number N (3, 3, 3, 3; 2)

An Upper Bound for the Turr an Number T 3 (n; 4)

Let t r (n; r + 1) denote the smallest integer m such that every r-uniform hypergraph on n vertices with m + 1 edges must contain a complete graph on r + 1 vertices. In this paper, we prove that lim n!1

The 3-colored Ramsey number for a 3-uniform loose path of length 3

The values of hypergraph 2-color Ramsey numbers for loose cycles and paths have already been determined. The only known value for more than 2 colors is R(C 3 ; 3) = 8, where C 3 3 is a 3-uniform loose cycle of length 3. Here we determine that R(P 3 3 ; 3) = 9, where P 3 3 is a 3-uniform loose path of length 3. Our proof relies on the determination of the Turán number ex3(9;P 3 3 ). We also find...

number 3

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On the Ramsey number R(3, 6)

The lower bound for the classical Ramsey number R(4, 6) is improved from 35 to 36. The author has found 37 new edge colorings of K35 that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found ...

3 A Randomized Upper Bound for