TY - JOUR

T1 - Monochromatic hilbert cubes and arithmetic progressions

AU - Balogh, József

AU - Lavrov, Mikhail

AU - Shakan, George

AU - Wagner, Adam Zsolt

N1 - Funding Information:
∗Research of the first author is partially supported by NSF Grant DMS-1500121 and Arnold O. Beckman Research Award (UIUC) Campus Research Board 18132 and the Langan Scholar Fund (UIUC).
Publisher Copyright:
© The authors.

PY - 2019

Y1 - 2019

N2 - 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.(forumala presented). We show the following relation between the Hilbert cube number and the Van der Waerden number. Let k > 3 be an integer. Then for every > 0, there is a c > 0 such that Thus we improve upon state of the art lower bounds for h(k; 4) conditional on W(k; 2) being signiffcantly larger than 2k. In the other direction, this shows that if the Hilbert cube number is close to 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 Z, one has (forumala presented).

AB - 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.(forumala presented). We show the following relation between the Hilbert cube number and the Van der Waerden number. Let k > 3 be an integer. Then for every > 0, there is a c > 0 such that Thus we improve upon state of the art lower bounds for h(k; 4) conditional on W(k; 2) being signiffcantly larger than 2k. In the other direction, this shows that if the Hilbert cube number is close to 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 Z, one has (forumala presented).

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U2 - 10.37236/7917

DO - 10.37236/7917

M3 - Article

AN - SCOPUS:85067287403

VL - 26

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

M1 - P2.22

ER -