A quasi-polynomial bound for the diameter of graphs of polyhedra

The diameter of the graph of a $d$-dimensional polyhedron with $n$ facets is at most $n^{\log d+2}$Comment: 2 page

On the average rank of LYM-sets

Let S be a finite set with some rank function r such that the Whitney numbers wi = |{x S|r(x) = i}| are log-concave. Given so that wk − 1 < wk wk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every F S with cardinality |F| W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xn m + 1

On a problem of Yuzvinsky on separating the n-cube

AbstractThe following problem of Yuzvinsky is solved here: how many vertices of the n-cube must be removed from it in order that no connected component of the rest contains an antipodal pair of vertices? Some further results and problems are described as well

On a conjecture of milner on k-graphs with non-disjoint edges

AbstractThe following theorem is proven. It is a slight generalization of a conjecture of Eric Milner.Consider two families, one consisting of k and the other of l element subsets of an n element set. Let each member of one have nonempty intersection with each member of the other and let k+l be less than or equal to n.Then either there are no more than ((n−1k−1)) members of the first family or there are fewer than ((n−1l−1)) members of the second

Semigroups and their topologies arising from Green's left quasiorder

[EN] Given a semigroup (S, ·), Green’s left quasiorder on S is given by a ≤ b if a = u · b for some u ϵ S1. We determine which topological spaces with five or fewer elements arise as the specialization topology from Green’s left quasiorder for an appropriate semigroup structure on the set. In the process, we exhibit semigroup structures that yield general classes of finite topological spaces, as well as general classes of topological spaces which cannot be derived from semigroup structures via Green’s left quasiorder.Richmond, B. (2008). Semigroups and their topologies arising from Green's left quasiorder. Applied General Topology. 9(2):143-168. doi:10.4995/agt.2008.1795.SWORD14316892Almeida, J. (2001). Some key problems on finite semigroups. Semigroup Forum, 64(2), 159-179. doi:10.1007/s002330010098Ern�, M., & Stege, K. (1991). Counting finite posets and topologies. Order, 8(3), 247-265. doi:10.1007/bf00383446Forsythe, G. E. (1955). SWAC Computes 126 Distinct Semigroups of Order 4. Proceedings of the American Mathematical Society, 6(3), 443. doi:10.2307/2032786Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M., & Scott, D. S. (2003). Continuous Lattices and Domains. doi:10.1017/cbo9780511542725P. A. Grillet, Semigroups, Marcel Dekker, New York. 1995.D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, The number of semigroups of order n, Proc. Am. Math. Soc. 55, no. 1 (1976), 227–232.Richmond, T. A. (1998). Quasiorders, principal topologies, and partially ordered partitions. International Journal of Mathematics and Mathematical Sciences, 21(2), 221-234. doi:10.1155/s0161171298000325Satoh, S., Yama, K., & Tokizawa, M. (1994). Semigroups of order 8. Semigroup Forum, 49(1), 7-29. doi:10.1007/bf02573467K. Tetsuya, T. Hashimoto, T. Akazawa, R. Shibata, T. Inui, and T. Tamura, All semigroups of order at most 5, J. Gakugei Tokushima Univ. Nat. Sci. Math. 6 (1955), 19–39

An upper bound for the Ramsey numbers r(K3,G)

AbstractThe Ramsey number r(H,G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph KN in red and blue, it must contain either a ŕed H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K3,G)⩽2q+1 where G has q edges. In other words, any graph on 2q+1 vertices with independence number at most 2 contains every (isolate-free) graph on q edges. This establishes a 1980 conjecture of Harary. The result is best possible as a function of q

On the Number of Latent Subsets of Intersecting Collections

Not AvailableSupported in part by O.N.R. Contract N00014-67-A-0204-0016 and supported in part by the U.S. Army Research Office (Durham) under contract DAHCO4-70-C-0058