Combinatorics of `unavoidable complexes'

Marija Jeli\'c Milutinovi\'c; Du\v{s}ko Joji\'c; Marinko; Timotijevi\'c; Sini\v{s}a T. Vre\'cica; Rade T. \v{Z}ivaljevi\'c

arXiv:1612.09487·math.CO·May 14, 2019·Eur. J. Comb.

Combinatorics of `unavoidable complexes'

Marija Jeli\'c Milutinovi\'c, Du\v{s}ko Joji\'c, Marinko, Timotijevi\'c, Sini\v{s}a T. Vre\'cica, Rade T. \v{Z}ivaljevi\'c

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TL;DR

This paper investigates the combinatorial properties of $r$-unavoidable simplicial complexes, which are characterized by their partition number, motivated by topological problems like Tverberg-type theorems.

Contribution

It introduces a detailed study of $r$-unavoidable complexes, connecting their combinatorics to topological problems and the constraint method used in recent research.

Findings

01

Characterization of $r$-unavoidable complexes

02

Connections to Tverberg-Van Kampen-Flores problems

03

Application of the constraint method in combinatorics

Abstract

The partition number $π (K)$ of a simplicial complex $K \subset 2^{[n]}$ is the minimum integer $ν$ such that for each partition $A_{1} ⊎ \dots ⊎ A_{ν} = [n]$ of $[n]$ at least one of the sets $A_{i}$ is in $K$ . A complex $K$ is $r$ -unavoidable if $π (K) \leq r$ . Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by the `constraint method' of Blagojevi\'{c}, Frick, and Ziegler, arXiv:1401.0690 [math.CO], we study the combinatorics of $r$ -unavoidable complexes.

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