This paper proves impossibility results on achievable parameters class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si1.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=192a5ab18adce2b778d6f07807cbf876" title="Click to view the MathML source">(α,β)class="mathContainer hidden">class="mathCode"> in the regime of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si6.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=171462487b5900edf6f3639ca9de83aa" title="Click to view the MathML source">n,k→∞class="mathContainer hidden">class="mathCode"> with a fixed ratio class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si352.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=b802b2eebcab5acb8d66b8463482d050">class="imgLazyJSB inlineImage" height="17" width="43" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0097316516300772-si352.gif">class="mathContainer hidden">class="mathCode">. This is done by developing a general criterion for existence of graph-homomorphism based on the semi-definite relaxation of the independence number of a graph (known as the Schrijver's θ-function). The criterion is then evaluated using some known and some new results from coding theory concerning the θ -function of Hamming graphs. As an example, it is shown that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si112.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=522846aaf5350e1e92ca3f73ff7f3605" title="Click to view the MathML source">β>1/2class="mathContainer hidden">class="mathCode"> and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si9.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=11c7f5963e5721e6cb3320db4d3f02af">class="imgLazyJSB inlineImage" height="16" width="11" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0097316516300772-si9.gif">class="mathContainer hidden">class="mathCode"> – integer, the class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si9.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=11c7f5963e5721e6cb3320db4d3f02af">class="imgLazyJSB inlineImage" height="16" width="11" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0097316516300772-si9.gif">class="mathContainer hidden">class="mathCode">-fold repetition map achieving class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si10.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=009b3b12fb5e784d65b46d6ad4df2ffb" title="Click to view the MathML source">α=βclass="mathContainer hidden">class="mathCode"> is asymptotically optimal.
Finally, constraints on configurations of points and hyperplanes in projective spaces over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si11.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=69a1a114ff17eca1376c34506c550557" title="Click to view the MathML source">F2class="mathContainer hidden">class="mathCode"> are derived.