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.
Categories: math.CO