Local k-colorings of graphs and hypergraphs
Journal of Combinatorial Theory Series B
Local and mean k-Ramsey numbers for complete graphs
Journal of Graph Theory
Properly colored subgraphs and rainbow subgraphs in edge-colorings with local constraints
Random Structures & Algorithms
On Pattern Ramsey Numbers of Graphs
Graphs and Combinatorics
An Upper Bound for Constrained Ramsey Numbers
Combinatorics, Probability and Computing
Mono-multi bipartite Ramsey numbers, designs, and matrices
Journal of Combinatorial Theory Series A
Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture
Journal of Graph Theory
Constrained Ramsey numbers of graphs
Journal of Graph Theory
Finding a monochromatic subgraph or a rainbow path
Journal of Graph Theory
Complete graphs with no rainbow path
Journal of Graph Theory
Hi-index | 0.00 |
For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.