Monochromatic Hamiltonian Berge-cycles in colored complete uniform hypergraphs

Abstract

We conjecture that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Bergecycle in every (r − 1)-coloring of the edges of K (r) n , the complete r-uniform hypergraph on n vertices. We prove the conjecture for r = 3, n 5 and its asymptotic version for r = 4. For general r we prove weaker forms of the conjecture: there is a Hamiltonian Berge-cycle in (r − 1)/2 -colorings of K (r) n for large n; and a Berge-cycle of order (1 − o(1))n in (r − log 2 r )-colorings of K (r) n . The asymptotic results are obtained with the Regularity Lemma via the existence of monochromatic connected almost perfect matchings in the multicolored shadow graph induced by the coloring of K (r) n .

Monochromatic cycle partitions of edge-colored graphs

“…We note here, again, that the use of a connected matching in this type of proof (first suggested by [24]) has become somewhat standard (see [5], [11], [14], [15], [12], [13] and [24]), so a proof of this lemma can be found in [11], for example. For the sake of completeness we repeat the proof.…”

“…Although the approach outlined above is now becoming 'standard', there are several technical solutions to handling 'almost complete' hypergraphs and their shadow graphs. We think that the following concept and the corresponding lemma (its straightforward proof is in [11]) are very convenient.…”

“…We start with a proposition (from [11]) about the connected components of a hypergraph. Proposition 1.6.…”

“…It turns out that the case t = 2 can be easily solved: for n > 4, R 2 (C n ) = n, i.e., there is a Hamiltonian Berge cycle in every 2-colouring of K (3) n (see [11]). In this paper we explore the 3-colour Ramsey number of a Berge cycle in 3-uniform hypergraphs.…”

Ramsey number of paths and connected matchings in Ore-type host graphs

“…[4], [9], [11], [12], [13], [14]. The crucial idea of this method is that "paths" in a statement to be proved are replaced by "connected matchings".…”