Monochromatic Hilbert Cubes and Arithmetic Progressions
نویسندگان
چکیده
منابع مشابه
A probabilistic threshold for monochromatic arithmetic progressions
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.
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Let N(k) = 2k/2k3/2f(k) and N(k) = 2k/2k1/2 g(k) where f(k) → ∞ and g(k) → 0 arbitrarily slowly as k → ∞. We show that the probability of a random 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic kterm arithmetic progression approaches 0, as k → ...
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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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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2019
ISSN: 1077-8926
DOI: 10.37236/7917