2024/09/18 by Fang, Yu, Wang, Xiaomiao, Feng, Tao
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2409.11907
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].