On Bollob\'as-type theorems of $d$-tuples
On Bollob\'as-type theorems of $d$-tuples
In 1965, Bollob\'as proved that for a Bollob\'as set-pair system $\{(A_i,B_i)\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i|+|B_i|}{A_i}^{-1}$ is $1$. In 2023, Heged\"{u}s and Frankl generalize the concept of Bollob\'as system to $d$-tuples, and made a conjecture: Suppose $\{(A_i^{(1)},\ldots,A_i^{(d)})\mid i\in[m]\}$ is a Bollob\'as system of $d$-tuples, the maximum value of $\sum_{i=1}^m\binom{|A_i^{(1)}|+\cdots+|A_i^{(d)}|}{|A_i^{(1)}|,\ldots,|A_i^{(d)}|}^{-1}$ …