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Colourings of integers and power-of-two differences (BxMO 2023, Problem 2)

Source: BxMO 2023, Problem 2

May 6, 2023

BxMOcombinatorics

Problem Statement

Determine all integers

k\geqslant 1

with the following property: given

k

different colours, if each integer is coloured in one of these

k

colours, then there must exist integers

a_1<a_2<\cdots<a_{2023}

of the same colour such that the differences

a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022}

are all powers of

2