Variations on Bollobás Systems of d d‐Partitions

This paper investigates five kinds of systems of ‐partitions of , 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 ‐partitions of 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 ‐partitions of .