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- W4300789439 abstract "The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic affine $k$--cube, that is, a set of the form$$left{x_0 + sum_{b in B} b : B subseteq Aright}$$ for some $|A|=k$ and $x_0 in mathbb{Z}$. We show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k geq 3$ be an integer. Then for every $epsilon >0$, there is a $c > 0$ such that $$h(k,4) ge min{W(lfloor c k^2rfloor, 2), 2^{k^{2.5-epsilon}}}.$$ Thus we improve upon state of the art lower bounds for $h(k,4)$ conditional on $W(k,2)$ being significantly larger than $2^k$. In the other direction, this shows that the if the Hilbert cube number is close its state of the art lower bounds, then $W(k,2)$ is at most doubly exponential in $k$. We also show the optimal result that for any Sidon set $A subset mathbb{Z}$, one has $$left|left{sum_{b in B} b : B subseteq Aright}right| = Omega( |A|^3) .$$" @default.
- W4300789439 created "2022-10-04" @default.
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- W4300789439 date "2018-05-22" @default.
- W4300789439 modified "2023-09-23" @default.
- W4300789439 title "Monochromatic Hilbert cubes and arithmetic progressions" @default.
- W4300789439 doi "https://doi.org/10.48550/arxiv.1805.08938" @default.
- W4300789439 hasPublicationYear "2018" @default.
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