The constructive lower-bound conjecture for Gallai–Ramsey numbers of hypergraphs

Let $H_1,H_2,\dots,H_t$ be $r$-uniform hypergraphs with $r>2$ and orders greater than $r$, where each $H_i$ is isomorphic either to a complete hypergraph or to a complete hypergraph with a single hyperedge removed. Let $gr(H_1,H_2,\dots,H_t;r)$ denote the corresponding Gallai–Ramsey number, and let $K_{r+1}^{(r)}$ be the complete $r$-uniform hypergraph on $r+1$ vertices. If $t\geq 1$, then

$$gr(H_1,H_2,\dots,H_t,K_{r+1}^{(r)},K_{r+1}^{(r)};r)\geq (gr(H_1,H_2,\dots,H_t;r)-1)^2+1.$$

The constructive lower-bound conjecture. The displayed inequality holds for all such $r$, $t$, and hypergraphs $H_1,\dots,H_t$. The preceding constructive theorems provide evidence for this claim, while proving it in general is difficult because the number of cases increases with the uniformity.