On Bollobás-type theorems of $d$-tuples
Erfei Yue
TL;DR
This work disproves the conjecture of Hegedűs and Frankl that every Bollobás system of $d$-tuples satisfies $\sum_{i=1}^m\binom{|A_i^{(1)}|+\cdots+|A_i^{(d)}|}{|A_i^{(1)}|,\ldots,|A_i^{(d)}|}^{-1}\le 1$ by providing a counterexample and establishing nonuniform upper bounds. Employing a probabilistic permutation method, it proves upper bounds for nonuniform and skew-nonuniform cases, with the $d=3$ bound $\le (n+3)/2$ and, for general $d$, the asymptotic bound $\le \frac{1}{d-1}\binom{n+d-2}{d-2}+O(n^{d-3})$, including a corresponding refinement for skew-nonuniform. In a separate exterior-algebra approach, it derives the uniform-space bound $m \le \binom{a_1+\cdots+a_d}{a_1,\ldots,a_d}$ for skew Bollobás systems of $d$-tuples of spaces, and deduces the uniform results on spaces (Theorem $\mathrm{Th:d-space}$ and $\mathrm{Th:d-skew-uniform}$). Overall, the results extend Bollobás-type theorems to $d$-tuples, clarifying nonuniform and uniform bounds and connecting to classical space- and matroid-type extensions.
Abstract
In 1965, Bollobás proved that for a Bollobás set-pair system $\{(A_i,B_i)\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i|+|B_i|}{A_i}^{-1}$ is $1$. Hegedüs and Frankl recently extended the concept of Bollobás systems to $d$-tuples, conjecturing that for a Bollobás system of $d$-tuples, $\{(A_i^{(1)},\ldots,A_i^{(d)})\mid i\in[m]\}$, 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 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.