Rainbow three-term progressions in ℤₙ with four or more colours

Abstract

For a surjective r-colouring of ℤₙ=ℤ/nℤ, call an ordered solution (x,y,z) of x+y=2z rainbow if its three coordinates carry three different colours. Elvin, Gonzales, Rodriguez and Wilbur determined the largest possible proportion of rainbow solutions for r=3 — it is 2/3, attained exactly when 3 | n — and asked what happens for r ≥ 4, where their counting argument gives only 1-1/r. We show that the whole gap at r=4 collapses to a single inequality about the colour classes: some pair of classes has size product at most the total defect Σᵢ|Aᵢ|²-M, where M counts the monochromatic solutions. That inequality implies the proportion 2/3 for every odd modulus, and we prove it unconditionally at every odd prime and at the composite moduli 9,15,21,27,33,39,51,57 and 69. The modulus 27 is the first for which the extension 0 → Zm 9 → Zm(27) → Zm3 → 0 does not split; we give an exact fibre decomposition of the defect twisted by the 2-cocycle of that extension and show the twist is not removable and is exactly what makes the modulus 27 work. Along the way we record an exact identity for odd n relating the rainbow count to the class sizes, two unconditional bounds strictly below 3/4, and the exact four-, five-, six- and seven-colour maxima at small moduli, from which the maxima appear to be 1-1/p(r) with p(r) the largest odd integer at most r — so that only an odd number of colours ever helps. All statements are machine-checked in Lean 4.

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