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">logk/loglogkan>an class="mathContainer hidden">an class="mathCode">ath altimg="si143.gif" overflow="scroll">athvariant="normal">logkalse">/athvariant="normal">logathvariant="normal">logkath>an>an>an>. Furthermore, we give bounds on n and k which imply that the van der Waerden complex is contractible.