Authors: Milutinović-Jelić, Marija
Jojić, Duško
Timotijević, Marinko
Vrećica, Siniša
Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Combinatorics of unavoidable complexes
Journal: European Journal of Combinatorics
Volume: 83
Issue Date: 1-Jan-2020
Rank: M22
ISSN: 0195-6698
DOI: 10.1016/j.ejc.2019.103004
The partition number π(K) of a simplicial complex K⊆2[n] is the minimum integer k such that for each partition A1⊎…⊎Ak=[n] of [n] at least one of the sets Ai is in K. A complex K is r-unavoidable if π(K)≤r. Simplicial complexes with small π(K) are important for applications of the “constraint method” (Blagojević et al., 2014) and serve as an input for the “index inequalities” (Jojić et al., 2018), such as (1.1). We introduce a “threshold characteristic” ρ(K) of K (Section 3) and define a fractional (linear programming) relaxation of π(K) (Section 4), which allows us to systematically generate interesting examples of r-unavoidable complexes and pave the way for new results of Van Kampen–Flores–Tverberg type.
Publisher: Elsevier
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Topology, geometry and global analysis on manifolds and discrete structures 

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