Variations on Bollob\'{a}s systems of $d$-partitions

TL;DR

This paper explores five types of Bollobás systems of $d$-partitions, unifies existing results, and disproves a conjecture for general systems while confirming it for strong systems.

Contribution

It unifies various Bollobás system variations and provides a counterexample to a conjecture, establishing its validity for strong systems.

Findings

Disproved a conjecture for general Bollobás systems

Confirmed the conjecture for strong Bollobás systems

Unified multiple variations of Bollobás systems

Abstract

This paper investigates five kinds of systems of $d$-partitions of $[n]$, including symmetric Bollob\'{a}s systems, strong Bollob\'{a}s systems, Bollob\'{a}s systems, skew Bollob\'{a}s systems, and weak Bollob\'{a}s systems. Many known results on variations of Bollob\'{a}s systems are unified. Especially we give a negative answer to a conjecture on Bollob\'{a}s systems of $d$-partitions of $[n]$ that was presented by Heged\"{u}s and Frankl [European J. Comb., 120 (2024), 103983]. Even though this conjecture does not hold for general Bollob\'{a}s systems, we show that it holds for strong Bollob\'{a}s systems of $d$-partitions of $[n]$.