A 4-uniform tight cycle is a hypergraph with cyclic ordering of its vertices such that edges are precisely the sets 4 consecutive in ordering. We prove Ramsey number for on 4n \((5 +o(1))n\). This asymptotically tight.

The Ramsey multiplicity R(G) of a graph G is the minimum number of monochromatic copies of G in any two-colouring of the edges of Kr(G), where r(G) denotes the Ramsey number of G. Here we prove that odd cycles have super-exponentially large Ramsey multiplicity: If Cn is an odd cycle of length n, then logR(Cn) = Θ(n logn).

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some result concerning both Ramsey saturated and unsaturated graph. In particular, we show that a cycle Cn and a Jahangir Jm Ramsey unsaturated or saturated graphs of R(Cn,Wm) and R(Pn, Jm), respectively. We also suggest an open problems.

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some conjectures and results concerning both Ramsey saturated and unsaturated graphs. In particular, we show that cycles Cn and paths Pn on n vertices are Ramsey unsaturated for all n ≥ 5. 1 Results and Conjectures Throughout this article, r...

An ordered graph G< is a graph G with vertices ordered by the linear ordering <. The ordered Ramsey number R(G<, c) is the minimum number N such that every ordered complete graph with c-colored edges and at least N vertices contains a monochromatic copy of G<. For unordered graphs it is known that Ramsey numbers of graphs with degrees bounded by a constant are linear with respect to the number ...

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp in [2], where it is shown that cycles (except for C4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle Cn, u...

We consider a hypergraph generalization of a conjecture of Burr and Erdős concerning the Ramsey number of graphs with bounded degree. It was shown by Chvátal, Rödl, Trotter, and Szemerédi [The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), no. 3, 239–243] that the Ramsey number R(G) of a graph G of bounded maximum degree is linear in |V (G)|. We derive...