{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:41:55Z","timestamp":1787323315453,"version":"build-2736575974"},"reference-count":10,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>For $S \\subseteq \\mathbb{Z}^+$ and k and r fixed positive integers, denote by $f(S,k;r)$ the least positive integer n (if it exists) such that within every r-coloring of $\\{1,2,\\dots,n\\}$ there must be a monochromatic sequence $\\{x_{1},x_{2},\\dots,x_{k}\\}$ with $x_{i}-x_{i-1} \\in S$ for $2 \\leq i \\leq k$. We consider the existence of $f(S,k;r)$ for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if S is an odd translate of the set of primes and $r=2$.<\/jats:p>","DOI":"10.1137\/s0895480103422196","type":"journal-article","created":{"date-parts":[[2007,9,28]],"date-time":"2007-09-28T18:12:32Z","timestamp":1191003152000},"page":"794-801","source":"Crossref","is-referenced-by-count":5,"title":["Avoiding Monochromatic Sequences With Special Gaps"],"prefix":"10.1137","volume":"21","author":[{"given":"Bruce M.","family":"Landman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aaron","family":"Robertson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,9,28]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"A. Balog,\n                      \n                        The prime\n                        k\n                        -tuplets conjecture on average\n                      \n                      , in Analytic Number Theory, Progr. Math. 85, Birkh\u00e4user, Boston, 1990, pp. 47\u201375.","DOI":"10.1007\/978-1-4612-3464-7_5"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1962-0148632-7"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(90)90021-N"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1999-003-9"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2004.10.006"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(92)90120-5"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(99)00137-5"},{"key":"R8","first-page":"173","volume":"52","author":"Landman B.","year":"1997","journal-title":"Util. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0315-3681","issn-type":"print"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"B. Landman and A. Robertson,\n                      Ramsey Theory on the Integers\n                      , Stud. Math. Libr. 24, AMS, Providence, RI, 2004.","DOI":"10.1090\/stml\/024"},{"key":"R10","first-page":"212","volume":"15","author":"van der Waerden B. L.","year":"1928","journal-title":"Nieuw Arch. Wisk.","ISSN":"https:\/\/id.crossref.org\/issn\/0028-9825","issn-type":"print"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895480103422196","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:55:00Z","timestamp":1787320500000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895480103422196"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/S0895480103422196"],"URL":"https:\/\/doi.org\/10.1137\/s0895480103422196","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}