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Long rainbow arithmetic progressions.
TL;DR: Conlon, Fox and Sudakov as discussed by the authors showed that T_k = O(k^2/log k) + O(1 + o(1+o(1)) log k) for every equinumerous color-coloring of $n\in \mathbb{N} for which there is a rainbow arithmetic progression.
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Abstract: Define $T_k$ as the minimal $t\in \mathbb{N}$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[tn]$ for all $n\in \mathbb{N}$. Jungic, Licht (Fox), Mahdian, Nesetril and Radoicic proved that $\lfloor{\frac{k^2}{4}\rfloor}\le T_k$. We almost close the gap between the upper and lower bounds by proving that $T_k \le k^2e^{(\ln\ln k)^2(1+o(1))}$. Conlon, Fox and Sudakov have independently shown a stronger statement that $T_k=O(k^2\log k)$.
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Short proofs of some extremal results III
TL;DR: This paper proves a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems, coming mainly from extremal graph theory and Ramsey theory.
References
Rainbow arithmetic progressions
Steve Butler,Craig Erickson,Leslie Hogben,Kirsten Hogenson,Lucas Kramer,Richard L. Kramer,Jephian C.-H. Lin,Ryan Martin,Derrick Stolee,Nathan Warnberg,Michael Young +10 more
TL;DR: In this article, the anti-van der Waerden properties of arithmetic progressions were investigated for positive integers n and k, and it was shown that the smallest number of colors with which the integers f1;:::;ng can be colored and still guarantee there is a rainbow arithmetic progression of length k was shown to be (log n) = (log k).
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Rainbow arithmetic progressions
Steve Butler,Craig Erickson,Leslie Hogben,Kirsten Hogenson,Lucas Kramer,Richard L. Kramer,Jephian C.-H. Lin,Ryan Martin,Derrick Stolee,Nathan Warnberg,Michael Young +10 more
TL;DR: In this paper, the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions were investigated and it was shown that for positive integers n and k, the smallest number of colors with which elements of the cyclic group of order can be colored and still guarantee there is a rainbow arithmetic progression of length n.
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A note on long rainbow arithmetic progressions
TL;DR: Jungi et al. as mentioned in this paper proved that for every k = O(k 5/2+ √ ϵ)-approximation using the K\H{o}v\'{a}ri-S\'{o]s-Tur\'{ a}n theorem and Wigert's bound on the divisor function.
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Independent arithmetic progressions
TL;DR: In this article, it was shown that there is a positive constant c such that any graph on vertex set [n] with at most cn^(2)/k^( 2) log k edges contains an independent set of order k whose vertices form an arithmetic progression.
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