Geodesic
A subexponential upper bound for van der Waerden numbers \(W(3,k)\)
The electronic journal of combinatorics, Tome 28 (2021) no. 2
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We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then $$N\le \exp(O(k^{1-c}))\,.$$
DOI : 10.37236/9704
Classification : 05D10, 11B25
Mots-clés : Szemerédi's theorem on arithmetic progressions, Roth's theorem 
@article{10_37236_9704,
     author = {Tomasz Schoen},
     title = {A subexponential upper bound for van der {Waerden} numbers {\(W(3,k)\)}},
     journal = {The electronic journal of combinatorics},
     year = {2021},
     volume = {28},
     number = {2},
     doi = {10.37236/9704},
     zbl = {1465.05181},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/9704/}
}
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Tomasz Schoen. A subexponential upper bound for van der Waerden numbers \(W(3,k)\). The electronic journal of combinatorics, Tome 28 (2021) no. 2. doi: 10.37236/9704

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