2019/05/09 by Balogh, József, Linz, William, Mattos, Letícia
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.03811
Define Tk as the minimal t∈ ℕ for which there is a rainbow arithmetic progression of length k in every equinumerous t-coloring of [tn] for all n∈ ℕ. Jungić, Licht (Fox), Mahdian, Nesetril and Radoicić proved that \lfloor(k2)/(4)\rfloor≤ Tk. We almost close the gap between the upper and lower bounds by proving that Tk ≤ k2e(lnln k)2(1+o(1)). Conlon, Fox and Sudakov have independently shown a stronger statement that Tk=O(k2log k).