A spectral sequence for the Hochschild cohomology of a coconnective dga

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence constructed by Cohen, Jones and Yan for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-p cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-p chains on its loop-space carries a theory of support varieties.Comment: Final version. The new version adds an application of the results to the construction of support varieties for modules over the chains algebra of certain loop-spaces. See Corollary 1.7 and Proposition 2.

Radon Numbers for Trees

Many interesting problems are obtained by attempting to generalize classical results on convexity in Euclidean spaces to other convexity spaces, in particular to convexity spaces on graphs. In this paper we consider $P_3$-convexity on graphs. A set $U$ of vertices in a graph $G$ is $P_3$-convex if every vertex not in $U$ has at most one neighbour in $U$. More specifically, we consider Radon numbers for $P_3$-convexity in trees. Tverberg's theorem states that every set of $(k-1)(d+1)-1$ points in $\mathbb{R}^d$ can be partitioned into $k$ sets with intersecting convex hulls. As a special case of Eckhoff's conjecture, we show that a similar result holds for $P_3$-convexity in trees. A set $U$ of vertices in a graph $G$ is called free, if no vertex of $G$ has more than one neighbour in $U$. We prove an inequality relating the Radon number for $P_3$-convexity in trees with the size of a maximal free set.Comment: 17 pages, 13 figure

Many HH-copies in graphs with a forbidden tree

For graphs $H$ and $F$, let $\operatorname{ex}(n, H, F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalises the well-studied Tur\'an numbers of graphs, was initiated recently by Alon and Shikhelman. We show that if $F$ is a tree then $\operatorname{ex}(n, H, F) = \Theta(n^r)$ for some integer $r = r(H, F)$, thus answering one of their questions.Comment: 9 pages, 1 figur