{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:25Z","timestamp":1753893865220,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$\u2013colored 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 $r$\u2013colored there exists a monochromatic affine $k$\u2013cube, that is, a set of the form$$\\left\\{x_0 + \\sum_{b \\in B} b : B \\subseteq A\\right\\}$$ for some $|A|=k$ and $x_0 \\in \\mathbb{Z}$.\r\nWe show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k \\geqslant 3$ be an integer. Then for every $\\epsilon &gt;0$, there is a $c &gt; 0$ such that $$h(k,4) \\geqslant \\min\\{W(\\lfloor c k^2\\rfloor, 2), 2^{k^{2.5-\\epsilon}}\\}.$$ 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 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 $k$.\r\nWe also show the optimal result that for any Sidon set $A \\subset \\mathbb{Z}$, one has $$\\left|\\left\\{\\sum_{b \\in B} b : B \\subseteq A\\right\\}\\right| = \\Omega( |A|^3) .$$<\/jats:p>","DOI":"10.37236\/7917","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:14:12Z","timestamp":1578640452000},"source":"Crossref","is-referenced-by-count":2,"title":["Monochromatic Hilbert Cubes and Arithmetic Progressions"],"prefix":"10.37236","volume":"26","author":[{"given":"J\u00f3zsef","family":"Balogh","sequence":"first","affiliation":[]},{"given":"Mikhail","family":"Lavrov","sequence":"additional","affiliation":[]},{"given":"George","family":"Shakan","sequence":"additional","affiliation":[]},{"given":"Adam Zsolt","family":"Wagner","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,5,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p22\/7835","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p22\/7835","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:14:19Z","timestamp":1579234459000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i2p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,17]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/7917","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,5,17]]},"article-number":"P2.22"}}