A note on a triangle-free — complete graph induced Ramsey number

A Note on Odd Cycle-Complete Graph Ramsey Numbers

The Ramsey number r(Cl,Kn) is the smallest positive integer m such that every graph of order m contains either cycle of length l or a set of n independent vertices. In this short note we slightly improve the best known upper bound on r(Cl,Kn) for odd l.

A note on the Ramsey number and the planar Ramsey number for C4 and complete graphs

We give a lower bound for the Ramsey number and the planar Ramsey number for C4 and complete graphs. We prove that the Ramsey number for C4 and K7 is 21 or 22. Moreover we prove that the planar Ramsey number for C4 and K6 is equal to 17.

Ramsey Number of a Connected Triangle Matching

We determine the 2-color Ramsey number of a connected triangle matching c(nK3) which is a graph with n vertex disjoint triangles plus (at least n − 1) further edges that keep these triangles connected. We obtain that R(c(nK3), c(nK3)) = 7n − 2, somewhat larger than in the classical result of Burr, Erdős and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ra...

The Universal Homogeneous Triangle-free Graph Has Finite Ramsey Degrees

The universal homogeneous triangle-free graph H3 has finite ‘big Ramsey degrees’: For each finite triangle-free graph G, there is a finite number n(G) such that for any coloring c of all copies of G in H3 into finitely many colors, there is a subgraph H′ of H3 which is isomorphic to H3 in which c takes on no more than n(G) many colors. The proof hinges on the following developments: a new flexi...

A Note on Triangle-Free Graphs