Let C(3,4, n) be the minimum number of four-element subsets (called blocks) of an n-element set, X, such that each three-element subset of X is contained in at least one block. Let L(3,4, n) = rn/4rn 1/3rn 2/2111. Schoenheim has shown that C(3,4, n) 2 L(3,4, n). The construction of Steiner quadruple systems of all orders n ~2 or 4 (mod 6) by Hanani (Canad. J. Math. 12 (1960), 145-157) can be us...

It is well known that, whenever k divides n, the complete k-uniform hypergraph on n vertices can be partitioned into disjoint perfect matchings. Equivalently, set of k-subsets an n-set parallel classes so that each class a partition n-set. This result as Baranyai's theorem, which guarantees existence Baranyai partitions. Unfortunately, proof theorem uses network flow arguments, making this none...

A Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. Part I shoewed there is a natural group action on Desc...

Abstract The five circles in the classical Pentagon theorem of Miquel are given as circumcircles certain triangles pentagon. If one chooses instead other triangles, gets a different configuration circles. This resulting carries three families concyclic quadruples points. Together with this gives total 20 radical axes each two these twenty all concurrent.