The Order of Long Rainbow Arithmetic Progressions
Prior state unknown→disproved
the true order is determined, and it is not the conjectured one
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combinatorics / Ramsey theory
The Order of Long Rainbow Arithmetic Progressions
Let $T_k$ be the least $t$ such that every equinumerous $t$-coloring of $[tn]$ contains a rainbow $k$-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured $T_k = \Theta(k^2)$; Conlon, Fox and Sudakov proved $T_k = O(k^2 \log k)$. The matching lower bound $T_k = \Omega(k^2 \log k)$ holds, so $T_k = \Theta(k^2 \log k)$ and the conjectured order is wrong.
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Let $T_k$ be the least $t$ such that every equinumerous $t$-coloring of $[tn]$ contains a rainbow $k$-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured $T_k = \Theta(k^2)$; Conlon, Fox and Sudakov proved $T_k = O(k^2 \log k)$. The matching lower bound $T_k = \Omega(k^2 \log k)$ holds, so $T_k = \Theta(k^2 \log k)$ and the conjectured order is wrong.
the true order is determined, and it is not the conjectured one
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