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# Alan Lew, Representability and boxicity of simplicial complexes

## Wednesday, June 16, 2021 @ 5:00 PM - 6:00 PM KST

Zoom ID: 869 4632 6610 (ibsdimag)

### Speaker

Alan Lew
Technion

An interval graph is the intersection graph of a family of intervals in the real line. Motivated by problems in ecology, Roberts defined the boxicity of a graph G to be the minimal k such that G can be written as the intersection of k interval graphs.

A natural higher-dimensional generalization of interval graphs is the class d-representable complexes. These are simplicial complexes that carry the information on the intersection patterns of a family of convex sets in $\mathbb R^d$. We define the d-boxicity of a simplicial complex X to be the minimal k such that X can be written as the intersection of k d-representable complexes.

A classical result of Roberts, later rediscovered by Witsenhausen, asserts that the boxicity of a graph with n vertices is at most n/2. Our main result is the following high dimensional extension of Roberts’ theorem: Let X be a simplicial complex on n vertices with minimal non-faces of dimension at most d. Then, the d-boxicity of X is at most $\frac{1}{d+1}\binom{n}{d}$.

Examples based on Steiner systems show that our result is sharp. The proofs combine geometric and topological ideas.

## Details

Date:
Wednesday, June 16, 2021
Time:
5:00 PM - 6:00 PM KST
Event Category:
Event Tags:

## Venue

Zoom ID: 869 4632 6610 (ibsdimag)

## Organizer

O-joung Kwon (권오정)
기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209