Linear threshold for non-list diversified tolerances with bounded colour-set size

Establish a minimum-degree threshold of order O(ℓε^{-1}) guaranteeing an α-majority edge colouring for every ε-excessive tolerance function on a colour set of size at most ℓ, for every fixed ℓ∈N and ε>0.

Background

Conjecture 25 is the non-list counterpart of Conjecture 23. Instead of assigning lists to edges, it assigns a global tolerance to each colour and assumes that the colour set has bounded size.

The paper notes that the best available non-list bounds follow from its list-based and uniform-vector results, but that achieving the conjectured linear dependence on ℓ and ε{-1} appears to require new methods.

References

Conjecture 25. There is a function 84 : N x (0, +00) > (l,E) > IR such that d4 = O(le-1) and for any fixed (l, E) E Nx (0, +00) and every graph G with the minimum degree d(G) ≥ 84(l, E) and any E-excessive tolerance function a : C > (0,1) with |C| < I there is an a-majority C-colouring of G.

— On list extensions of the majority edge colourings  (2502.12688 - Pękała et al., 18 Feb 2025) in Conjecture 25, Section 7 (Concluding remarks)