combinatorics / Additive Combinatorics, Ramsey Theory
Deng, Tidor and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no symmetrically coloured 4-term arithmetic progression, giving an $O(N^{\log_{22}3})$ coloring. The paper gives an $O_k(N^{4/k^2})$ coloring of $[N]$ avoiding symmetrically coloured $k$-APs for every even $k\ge4$, improving the exponent.
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Append-only history
Improves the exponent rather than answering the asked question: whether an $N^{o(1)}$ colouring exists remains open.
Research memory
Deng, Tidor and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no symmetrically coloured 4-term arithmetic progression, giving an $O(N^{\log_{22}3})$ coloring. The paper gives an $O_k(N^{4/k^2})$ coloring of $[N]$ avoiding symmetrically coloured $k$-APs for every even $k\ge4$, improving the exponent.
Improves the exponent rather than answering the asked question: whether an $N^{o(1)}$ colouring exists remains open.
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