范德华登数的新下界

IF 2.8 1区数学 Q1 MATHEMATICS

Forum of Mathematics Pi Pub Date : 2021-02-02 DOI:10.1017/fmp.2022.12

B. Green

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引用次数: 0

摘要

摘要证明了$[N]$的红-蓝着色性，它没有蓝色的3项等差数列，也没有长度为$e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$的红色等差数列。因此，双色范德华登数$w(3,k)$以$k^{b(k)}$为界，其中$b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$。在此之前，有数据支持的推测是$w(3,k) = O(k^2)$。

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New lower bounds for van der Waerden numbers

Abstract We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.

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来源期刊

Forum of Mathematics Pi Mathematics-Statistics and Probability

CiteScore

3.50

自引率

0.00%

发文量

审稿时长

19 weeks

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