2017/10/06 by Alfaro, Carlos A., Vázquez-Ávila, Adrián
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1710.02501
A linear system is a pair (P,L) where L is a family of subsets on a ground finite set P such that |l∩ l^′|≤ 1, for every l,l^′ ∈ L. If all elements of L of a linear system (P,L), then the linear system is called r-uniform linear system. The transversal number of a linear system (P,L), τ(P,L), is the minimum cardinality of a subset P⊆ P satisfying l∩P≠∅, for every l\inL. The 2-packing number of a linear system (P,L), ν2(P,L), is the maximum cardinality of a subset R\subseteqL such that, any three elements of R don't have a common point (are triplewise disjoint), that is, if three elements are chosen in R, then they are not incidents in a common point. For r≥2, let (P,L) be an r-uniform linear system. In "\sc M. A. Henning and A. Yeo: \it Hypergraphs with large transversal number, Discrete Math. \bf 313 (2013), no. 8, 959--966." Henning and Yeo state the following question: Is it true that if (P,L) is an r-uniform linear system then τ(P,L)≤(|P|+|L|)/(r+1) holds for all r≥2?. In this note, we give some results of r-uniform linear systems, whose 2-packing number is fixed, satisfying the inequality.