Document Zbl 0698.52005 - zbMATH Open
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Topological methods in combinatorial problems. (English. Russian original) Zbl 0698.52005
The author presents a survey of results on a topic that is close to the Borsuk-Ulam theorem on antipodes. He analyzes developments in the Hadwiger conjecture: If \(m+1\) closed sets cover the sphere \(S^ m\) then at least one element of the covering contains pairs of points that realize any angle \(\phi\), \(0\leq \phi \leq \pi\). He shows that if \(r\) is the greatest integer such that any closed covering with \(r\) sets of \(S^ m\) contains an element that contains pairs of points realizing any angle from 0 to \(\pi\), then \(m\leq r\leq m+1\).
The author formulates a number of conjectures that are connected with this conjecture and the Kneser conjecture. Here is one such conjecture: If all \(n\)-point subsets of an \([(l-1)(q-1)+\ln]\)-point set are partitioned into \(q\) classes, then in at least one class there will be \(l\) pairwise disjoint \(n\)-point subsets.