64 research outputs found
Spherical two-distance sets
A set S of unit vectors in n-dimensional Euclidean space is called spherical
two-distance set, if there are two numbers a and b, and inner products of
distinct vectors of S are either a or b. The largest cardinality g(n) of
spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known
to be tight for n=2,6,22. The set of mid-points of the edges of a regular
simplex gives the lower bound L(n)=n(n+1)/2 for g(n.
In this paper using the so-called polynomial method it is proved that for
nonnegative a+b the largest cardinality of S is not greater than L(n). For the
case a+b<0 we propose upper bounds on |S| which are based on Delsarte's method.
Using this we show that g(n)=L(n) for 6<n<22, 23<n<40, and g(23)=276 or 277.Comment: 9 pages, (v2) several small changes and corrections suggested by
referees, accepted in Journal of Combinatorial Theory, Series
KKM type theorems with boundary conditions
We consider generalizations of Gale's colored KKM lemma and Shapley's KKMS
theorem. It is shown that spaces and covers can be much more general and the
boundary KKM rules can be substituted by more weaker boundary assumptions.Comment: 13 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1406.6672 by other author
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