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