Gallai–Ramsey numbers for 3-uniform rainbow Berge triangles and monochromatic linear paths or cycles

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 ...

For an arbitrary graph G, a hypergraph H is called Berge-G if there is an injection i : V ( G ) ⟶ V ( H ) and a bijection ψ : E ( G ) ⟶ E ( H ) such that for each e = u v ∈ E ( G ), we have { i ( u ) , i ( v ) } ⊆ ψ ( e ). We denote by B r G, the ...

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{}}$