Lopsided sets and orthant-intersection by convex sets

Abstract.To a large extent the present work is far from being conclusive, instead, new directions of research in combinatorial extremal theory are started. Also questions concerning generalizations are immediately noticeable.The incentive came from problems in several fields such as Algebra, Geometry, Probability, Information and Complexity Theory. Like several basic combinatorial problems they may play a role in other fields. For scenarios of interplay we refer also to [9].

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“…Lopsided sets where introduced in [10] in the study of convex sets. There are several equivalent definitions [11].…”

Section: E Results For Lopsided Sets In Combinatorial Languagementioning

confidence: 99%

On attractive and friendly sets in sequence spaces

Ahlswede

Khachatrian²

2005

Electronic Notes in Discrete Mathematics

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“…For lopsided sets the (i)⇔(ii) part of the corollary corresponds to a characterization due to Lawrence [30]. For the complete statement denote −− := {Q −− n | n ≥ 3}.…”

Section: Further Characterizations and Recognition Of Tope Graphsmentioning

confidence: 99%

On Tope Graphs of Complexes of Oriented Matroids

Knauer

Marc

2019

Discrete Comput Geom

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We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs.Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Moreover, we obtain purely graph theoretic polynomial time recognition algorithms for tope graphs of the above and a finite list of excluded partial cube minors for the bounded rank case.In particular, our results answer a relatively long-standing open question in oriented matroids and can be seen as identifying the theory of (complexes of) oriented matroids as a part of metric graph theory. Another consequence is that all finite Pasch graphs are tope graphs of complexes of oriented matroids, which confirms a conjecture of Chepoi and the two authors.

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“…While

V C ‐ dim (scriptS) \leq \log n

always holds, for some concept classes

normalV normalC ‐ dim (S)

is much smaller than

\log n

and thus the inequality

(∣ E ∣ / ∣ V ∣) \leq V C ‐ dim (scriptS)

presents a significant improvement over the folklore inequality

∣ E ∣ / ∣ V ∣ \leq \log n

. This is the case of maximum concept classes and, more generally, of lopsided systems . In this case,

normalV normalC ‐ dim (S)

is exactly the dimension of the largest subcube of

G (scriptS)

and this dimension may not depend at all on the number of sets of

scriptS

.…”

Section: Introductionmentioning

confidence: 99%

On density of subgraphs of Cartesian products

Chepoi

Labourel

Ratel

2019

Journal of Graph Theory

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In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs G of Cartesian products G1□⋯□Gm of arbitrary connected graphs. Namely, we show that (∣E(G)false|/∣V(G)∣)≤⌈2max{dens(G1),…,dens(Gm)}⌉log∣V(G)∣, where dens(H) is the maximum ratio ∣E(H′)∣/∣V(H′)∣ taken over all subgraphs H′ of H. We introduce the notions of VC‐dimension normalVnormalC‐dim⁡(G) and VC‐density normalVnormalCnormal‐normaldnormalenormalnnormals(G) of a subgraph G of a Cartesian product G1□⋯□Gm, generalizing the classical Vapnik‐Chervonenkis dimension of set‐families (viewed as subgraphs of hypercubes). We prove that if G1,…,Gm belong to the class G(H) of all finite connected graphs not containing a given graph H as a minor, then for any subgraph G of G1□⋯□Gm the sharper inequality (∣E(G)∣/∣V(G)∣)≤μ(H)⋅VC‐dim⁡*(G) holds, where μ(H) is the supremum of the densities of the graphs from G(H). We refine and sharpen these two results to several specific graph classes. We also derive upper bounds (some of them polylogarithmic) for the size of adjacency labeling schemes of subgraphs of Cartesian products.

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Lopsided sets and orthant-intersection by convex sets

Cited by 46 publications

References 5 publications

On attractive and friendly sets in sequence spaces

On attractive and friendly sets in sequence spaces

On Tope Graphs of Complexes of Oriented Matroids

On density of subgraphs of Cartesian products

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