Variations on Bollobás systems of $d$-partitions

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