A probabilistic threshold for monochromatic arithmetic progressions
نویسنده
چکیده
We show that √ k2k/2 is, roughly, the threshold where, under mild conditions, on one side almost every coloring contains a monochromatic k-term arithmetic progression, while on the other side, there are almost no such colorings.
منابع مشابه
Monochromatic 4-term arithmetic progressions in 2-colorings of Zn
This paper is motivated by a recent result of Wolf [12] on the minimum number of monochromatic 4-term arithmetic progressions (4-APs, for short) in Zp, where p is a prime number. Wolf proved that there is a 2-coloring of Zp with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic and non-constructive. In this paper, we present an explicit and simple construct...
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{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...
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Thus if all primes are colored with k colors, then there exist arbitrarily long monochromatic arithmetic progressions. This is a van der Waerden-type [9] theorem for primes. (The well-known van der Waerden theorem states that for any k-coloring of all positive integers, there exist arbitrarily long monochromatic arithmetic progressions.) On the other hand, Schur’s theorem [7] is another famous ...
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عنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 137 شماره
صفحات -
تاریخ انتشار 2016