18,053 research outputs found
Bounding Helly numbers via Betti numbers
We show that very weak topological assumptions are enough to ensure the
existence of a Helly-type theorem. More precisely, we show that for any
non-negative integers and there exists an integer such that
the following holds. If is a finite family of subsets of such that for any
and every
then has Helly number at most . Here
denotes the reduced -Betti numbers (with singular homology). These
topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based
approach to build, given an arbitrary simplicial complex , some well-behaved
chain map .Comment: 29 pages, 8 figure
The Goldman bracket determines intersection numbers for surfaces and orbifolds
In the mid eighties Goldman proved an embedded curve could be isotoped to not
intersect a closed geodesic if and only if their Lie bracket (as defined in
that work) vanished. Goldman asked for a topological proof and about extensions
of the conclusion to curves with self-intersection. Turaev, in the late
eighties, asked about characterizing simple closed curves algebraically, in
terms of the same Lie structure. We show how the Goldman bracket answers these
questions for all finite type surfaces. In fact we count self-intersection
numbers and mutual intersection numbers for all finite type orientable
orbifolds in terms of a new Lie bracket operation, extending Goldman's. The
arguments are purely topological, or based on elementary ideas from hyperbolic
geometry.
These results are intended to be used to recognize hyperbolic and Seifert
vertices and the gluing graph in the geometrization of three manifolds. The
recognition is based on the structure of the String Topology bracket of three
manifolds
Restricted frame graphs and a conjecture of Scott
Scott proved in 1997 that for any tree , every graph with bounded clique
number which does not contain any subdivision of as an induced subgraph has
bounded chromatic number. Scott also conjectured that the same should hold if
is replaced by any graph . Pawlik et al. recently constructed a family
of triangle-free intersection graphs of segments in the plane with unbounded
chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This
shows that Scott's conjecture is false whenever is obtained from a
non-planar graph by subdividing every edge at least once.
It remains interesting to decide which graphs satisfy Scott's conjecture
and which do not. In this paper, we study the construction of Pawlik et al. in
more details to extract more counterexamples to Scott's conjecture. For
example, we show that Scott's conjecture is false for any graph obtained from
by subdividing every edge at least once. We also prove that if is a
2-connected multigraph with no vertex contained in every cycle of , then any
graph obtained from by subdividing every edge at least twice is a
counterexample to Scott's conjecture.Comment: 21 pages, 8 figures - Revised version (note that we moved some of our
results to an appendix
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