On Bollobás-type theorems of $d$-tuples

In 1965, Bollobás proved that for a Bollobás set-pair system $\{(A_i,B_i)\mid i\in [m]\}$, the maximum value of ${\sum }_{i=1}^m{\left(\genfrac{}{}{0px}{}{\mathrm{\mid }{A}_i\mathrm{\mid }+\mathrm{\mid }{B}_i\mathrm{\mid }}{A_i}\right)}^{-1}$ is $1$. Hegedüs and Frankl recently extended the concept of Bollobás systems to $d$-tuples, conjecturing that for a Bollobás system of $d$-tuples, $\{(A_i^{(1)},\dots ,A_i^{(d)})\mid i\in [m]\}$, the maximum value of ${\sum }_{i=1}^m{\left(\genfrac{}{}{0px}{}{\mathrm{\mid }{A}_i^{(1)}\mathrm{\mid }+\cdots +\mathrm{\mid }{A}_i^{(d)}\mathrm{\mid }}{\mathrm{\mid }{A}_i^{(1)}\mathrm{\mid },\dots ,\mathrm{\mid }{A}_i^{(d)}\mathrm{\mid }}\right)}^{-1}$ 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 Bollobás systems of $d$-tuples in Hegedüs and Frankl’s paper. Finally, we determine the maximum size of a uniform skew Bollobás system of $d$-tuples on both sets and spaces.