A Short Proof of Kneser's Conjecture
نویسنده
چکیده
In the paper a short proof is given for Kneser's conjecture. The proof is based on Borsuk's theorem and on a theorem of Gale. Recently, LOV~SZ has given a proof for Kneser's conjecture [4]. He used Borsuk's theorem. This fact gave the author the idea of the proof we present here. THEOREM (Kneser's conjecture [3]). If the n-tuples of a set of 2n + k elements are partitioned into k + 1 classes, then one of the classes contains two disjoint n-tuples. For the graph-theoretic formulation of this theorem and other comments on it see Lovrisz's paper [4]. We will need two theorems. Put S, = (x E Rk+l: I/ x // = 1) and H(a) = BORSUK'S THEOREM. If Sk is the union of k + 1 sets which are open in Sk , then one of these sets contains antipodalpoints. GALE'S THEOREM. Zf n and k are nonnegative integers, then there is a set V C Sk with 2n + k elements such that / H(a) n V / 3 n for each a E Sk. In the original form of Borsuk's theorem the k + 1 sets are supposed to be closed in Sk (see [l]). It is an easy exercise to show from the original form that the above statement is true. For a proof of Gale's theorem see [2]. It is perhaps worth mentioning that this theorem is the dual (in the sense of Gale diagrams) of the fact that there exist in Rd [d/2]-neighborly polytopes with any number of vertices.
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عنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 25 شماره
صفحات -
تاریخ انتشار 1978