4 research outputs found
Finitely forcible graphons with an almost arbitrary structure
Graphons are analytic objects representing convergent sequences of large graphs. A graphon is said to be finitely forcible if it is determined by finitely many subgraph densities, i.e., if the asymptotic structure of graphs represented by such a graphon depends only on finitely many density constraints. Such graphons appear in various scenarios, particularly in extremal combinatorics.
Lovasz and Szegedy conjectured that all finitely forcible graphons possess a simple structure. This was disproved in a strong sense by Cooper, Kral and Martins, who showed that any graphon is a subgraphon of a finitely forcible graphon. We strenghten this result by showing for every ε>0 that any graphon spans a 1−ε proportion of a finitely forcible graphon
Finitely forcible graphons with an almost arbitrary structure
Graphons are analytic objects representing convergent sequences of large
graphs. A graphon is said to be finitely forcible if it is determined by
finitely many subgraph densities, i.e., if the asymptotic structure of graphs
represented by such a graphon depends only on finitely many density
constraints. Such graphons appear in various scenarios, particularly in
extremal combinatorics.
Lovasz and Szegedy conjectured that all finitely forcible graphons possess a
simple structure. This was disproved in a strong sense by Cooper, Kral and
Martins, who showed that any graphon is a subgraphon of a finitely forcible
graphon. We strenghten this result by showing for every that
any graphon spans a proportion of a finitely forcible graphon
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