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arXiv · 2409.11907

Variations on Bollob\'{a}s systems of $d$-partitions

Yu Fang·Xiaomiao Wang·Tao Feng
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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 $[n]$.

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Yu Fang, Xiaomiao Wang, Tao Feng. 2024-09-18. Variations on Bollob\'{a}s systems of $d$-partitions. https://arxiv.org/abs/2409.11907

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