A note on the maximum size of the ground set of skew Bollobás systems

• Published in: Discrete mathematics Vol. 348; no. 12; p. 114650
• Main Authors: Fang, Yu, Wang, Xiaomiao, Feng, Tao
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2025
• Subjects: Bollobás system; d-partition; Skew Bollobás system; Weak Bollobás system; Weak Bollobás system; Skew Bollobás system; Bollobás system; d-partition
• ISSN: 0012-365X
• DOI: 10.1016/j.disc.2025.114650

A skew Bollobás system D={(Ai(1),…,Ai(d)):1≤i≤m} is a collection of d pairwise disjoint subsets of [n] such that for any 1≤i<j≤m, there exist 1≤p<q≤d with Ai(p)∩Aj(q)≠∅. Denote by nskew(a1,…,ad) the maximum size of the ground set ⋃i=1m⋃r=1dAi(r) of a skew Bollobás system D such that |Ai(r)|≤ar for i∈[m] and r∈[d]. We show that for any positive integers a1,…,ad,nskew(a1,…,ad)=∑j=1∑r=1dar−1∑0≤λr≤ar,∀r∈[d]λ1+⋯+λd=j(∑r=1dar−ja1−λ1,…,ad−λd)+1. In particular for d=2, we havenskew(a1,a2)=(a1+a2+2a1+1)−(a1+a2a1)−1 for any non-negative integers a1 and a2. This corrects a typo in the upper bound on nskew(a1,a2) given by Nagy and Patkós (2015) [7].