Grinshpun–Sárközy conjecture on tiling bounded-degree graph sequences

Let \mathcal{F}={F_1,F_2,\dots\} be a graph sequence with $v(F_i)=i$ and maximum degree $\Delta(F_i)\leq\Delta$ for all $i$. For an $r$-edge coloured complete graph, let $\tau_r(\mathcal{F})$ be the least $s$ such that its vertex set has a monochromatic $\mathcal{F}$-tiling of size at most $s$, where a monochromatic $\mathcal{F}$-tiling consists of monochromatic subgraphs, each isomorphic to an element of $\mathcal{F}$, whose vertex sets partition the complete graph's vertex set.

Grinshpun–Sárközy conjecture. For every positive integer $r$ there exists a constant $C_r$ such that, for every $\Delta\geq 2$ and every $\Delta$-bounded graph sequence $\mathcal{F}$,

$$\tau_r(\mathcal{F})\leq \exp(\Delta^{C_r}).$$

The conjecture concerns the finiteness and quantitative growth of the tiling number for more than two colours. Corsten and Mendonça later proved finiteness with a triple-exponential bound, and the paper proves the stronger bound $\tau_r(\mathcal{F})\leq\exp(C_r\Delta)$ for bipartite bounded-degree graph sequences, but the full conjecture for general sequences is not resolved here.