Monochromatic Hamiltonian 3‐tight Berge cycles in 2‐colored 4‐uniform hypergraphs

Abstract

A Gallai-coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Gallai-colorings occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper) or information theory. Gallai-colorings extend 2-colorings of the edges of complete graphs. They actually turn out to be close to

Dirac-Type Questions For Hypergraphs — A Survey (Or More Problems For Endre To Solve)

“…In the smallest case of 8 = 4, that is, the unique 3-graph with 4 vertices and 2 edges, they show that (10) This seems surprising, since the obtained threshold is about twice smaller than the threshold for perfect matchings. For 8 :2: 6, the value of P2(3, n; C~3,1)) remains unknown , except for large 8 when P2(3, n; C~3,1)) rv n/4, but unlike in (10), here the asymptotics is also as 8 ---t 00 (see Theorem 1.2 in [17]). …”

Stability of the path–path Ramsey number

“…Actually, in the application [6] we needed a slightly stronger statement, namely that for each vertex of P we have at least (c/2) log n choices, but this is a straightforward corollary of the proof. …”

“…Since this is what we needed in a recent application [6], actually we prove the result in a slightly more general context; we work with 2-edge multicolorings (G 1 , G 2 ) of a graph G. Here multicoloring means that the edges can receive more than one color, i.e. the graphs G i are not necessarily edge disjoint.…”

Monochromatic Hamiltonian Berge-cycles in colored hypergraphs

“…For small cases: The case c = 2, t = 3 and r = 4 of Conjecture 2 was proved in [5]. In [3] Conjecture 1 was proved for r = 3 and an asymptotic result on this conjecture for r = 4 was obtained using the method of Regularity Lemma.…”