On Bollob\'as-type theorems of $d$-tuples

Abstract

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 $1$. Heged\"{u}s and Frankl recently extended the concept of Bollob\'as systems to $d$-tuples, conjecturing that for a Bollob\'as system of $d$-tuples, $\{(A_i^{(1)},\ldots,A_i^{(d)})\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i^{(1)}|+\cdots+|A_i^{(d)}|}{|A_i^{(1)}|,\ldots,|A_i^{(d)}|}^{-1}$ is also $1$. This paper refutes this conjecture and establishes an upper bound for the sum. In the case $d=3$, the derived upper bound is asymptotically tight. Furthermore, we sharpen an inequality for skew Bollob\'as systems of $d$-tuples in Heged\"{u}s and Frankl's paper. Finally, we determine the maximum size of a uniform skew Bollob\'as system of $d$-tuples on both sets and spaces.