Abstract: Sidorenko’s conjecture states that the number of homomorphisms from a bipartite graph H to a graph G is at least the expected number of homomorphisms from H to the binomial random graph with the same expected edge density as G. In this talk, I will present two approaches to the conjecture. First, I will introduce the notion of tree-arrangeability, where a bipartite graph H with bipart...

Denote by r(n) the length of a shortest integer sequence on a circle containing all permutations of the set {1, 2, . . . , n} as subsequences. Hansraj Gupta conjectured in 1981 that r(n) 6 n 2 2 . In this paper, we confirm the conjecture for the case where n is even and show that r(n) < n 2 2 + n 4 − 1 if n is odd. © 2012 Elsevier B.V. All rights reserved.

We prove that Conjecture H of Caporaso et al. [CHarM], §6 together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHarM]; a few applications in arithmetic and geometry are stated. As in [א], one uses the fact that rational points on X B are n-tuples of points when trying to bound n, as well as the fact that the Noetherian inductio...

The square G2 of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Let χ(H) and χl(H) be the chromatic number and the list chromatic number of H, respectively. A graph H is called chromatic-choosable if χl(H) = χ(H). It is an interesting problem to find graphs that are chromatic-choosable. Motivated by th...

in the present paper, among other results, a decomposition formula is given for the w-bounded continuous negative definite functions of a topological *-semigroup s with a weight function w into a proper h*-algebra a in terms of w-bounded continuous positive definite a-valued functions on s. a generalization of a well-known result of k. harzallah is obtained. an earlier conjecture of the author ...