Determine the minimum-degree threshold δ_{r,q} for small numbers of colors

Determine δ_{r,q} for all integers r ≥ 3 and q ≤ (r choose 2), where δ_{r,q} is the infimum δ > 0 such that there exists ζ > 0 with the property that for all sufficiently large n divisible by r, every n-vertex graph G with minimum degree at least δn and every q-edge-coloring of E(G) contains a K_r-tiling whose discrepancy is at least ζn (as defined in Definition 1.2).

Background

The paper studies the function δ{r,q}, the minimum degree threshold ensuring that every q-edge-coloring of an n-vertex graph (with n divisible by r) contains a K_r-tiling with linear discrepancy. It fully determines δ{r,q} for all sufficiently large q (namely q ≥ (r choose 2)) and recovers the known two-color case δ_{r,2} = r/(r+1).

The results exhibit a phase transition in q at (r+1 choose 2), and also settle many cases when q < (r choose 2) under a divisibility/arithmetic condition, showing tightness of the upper bound r/(r+1) there. The unresolved regime concerns the remaining small-color cases q ≤ (r choose 2) not covered by their arithmetic condition.

Consequently, the authors highlight as the main unresolved direction the complete determination of δ_{r,q} for all r ≥ 3 and q ≤ (r choose 2).