Variations on Bollobás systems of $d$-partitions

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 $[n]$.