Gallai–Ramsey Numbers for Paths

Given k graphs H 1 , H 2 , … , H k, the k-color Ramsey number R ( H 1 , H 2 , … , H k ) is defined as the minimum integer n such that every k-edge-coloring of K n contains a monochromatic H i, for some i ∈ [ k ]. If H 1 = H 2 = … = H k ...

The Gallai–Ramsey number $g{r}_k(K_3:H_1,H_2,\cdots ,H_k)$ is the smallest integer n such that every k-edge-colored $K_n$ contains either a rainbow $K_3$ or a monochromatic $H_i$ in color i for some $i\in [k]$. We define the star-critical Gallai–Ramsey number $g{r}_k^{\ast }(K_3:H_1,H_2,\cdots ,{\mathrm{...}}_{}$${}_{}{}_{\mathrm{}\mathrm{}}$${}_{\mathrm{}}$${}_{}$$$${}_{\mathrm{}}{}_{}$${}_{}^{}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{}$${}_{}^{}{}_{\mathrm{}}$${}_{}^{}{}_{\mathrm{}}$

Given two non-empty t-uniform hypergraphs G ( t ) , H ( t ) and a positive integer k, the hypergraph Gallai–Ramsey number gr k G ( t ) : H ( t ) is defined as the minimum positive integer N such that for all n ≥ N, every k-hyperedge-coloring of t-...