The van der Waerden complex

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• 作者：Richard Ehrenborg^a ; ^{richard.ehrenborg@uky.edu" class="auth_mail" title="E-mail the corresponding author} ; Likith Govindaiah^b ; ^{likithg@princeton.edu" class="auth_mail" title="E-mail the corresponding author} ; Peter S. Park^b ; ^{pspark@math.princeton.edu" class="auth_mail" title="E-mail the corresponding author} ; Margaret Readdy^a ; ^{margaret.readdy@uky.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Van der Waerden complex ; Discrete Morse theory ; Arithmetic progressions
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：172
• 期：Complete
• 页码：287-300
• 全文大小：343 K

文摘

We introduce the van der Waerden complex an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302207&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302207&_rdoc=1&_issn=0022314X&md5=408122782a43625f2e4a52aaceedd754" title="Click to view the MathML source">vdW(n,k)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="normal">vdWalse">(n,kalse">)ath>an>an>an> defined as the simplicial complex whose facets correspond to arithmetic progressions of length k   in the vertex set an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302207&_mathId=si2.gif&_user=111111111&_pii=S0022314X16302207&_rdoc=1&_issn=0022314X&md5=0859c0bd7bc9fc7b69c1162acb7d2385" title="Click to view the MathML source">{1,2,…,n}an>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">alse">{1,2,…,nalse">}ath>an>an>an>. We show the van der Waerden complex an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302207&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302207&_rdoc=1&_issn=0022314X&md5=408122782a43625f2e4a52aaceedd754" title="Click to view the MathML source">vdW(n,k)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">athvariant="normal">vdWalse">(n,kalse">)ath>an>an>an> is homotopy equivalent to a CW  -complex whose cells asymptotically have dimension at most an id="mmlsi143" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302207&_mathId=si143.gif&_user=111111111&_pii=S0022314X16302207&_rdoc=1&_issn=0022314X&md5=b46b43e3902db92b6a06126fa49361a4" title="Click to view the MathML source">log⁡k/log⁡log⁡kan>an class="mathContainer hidden">an class="mathCode">ath altimg="si143.gif" overflow="scroll">athvariant="normal">log⁡kalse">/athvariant="normal">log⁡athvariant="normal">log⁡kath>an>an>an>. Furthermore, we give bounds on n and k which imply that the van der Waerden complex is contractible.