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Omni-MATH / Determine all integers k≥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 …

Problem

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problem

Determine all integers k≥1k\geqslant 1 with the following property: given kk different colours, if each integer is coloured in one of these kk colours, then there must exist integers a1<a2<⋯<a2023a_1<a_2<\cdots<a_{2023} of the same colour such that the differences a2−a1,a3−a2,…,a2023−a2022a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022} are all powers of 22.
Plain-text mathematical notation (without MathML)
Determine all integers k≥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 a₁<a₂<⋯<a₂₀₂₃ of the same colour such that the differences a₂−a₁,a₃−a₂,…,a₂₀₂₃−a₂₀₂₂ are all powers of 2.
Original LaTeX notation
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 $2$.

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Source and history

task_id
a113762e-15d2-5c83-8cb4-b53f6d7f7d73
task_key
test--a113762e-15d2-5c83-8cb4-b53f6d7f7d73
task_revision_id
1
config / split
/ test

Official source

initial import