A constructive proof of the Bollobás–Varopoulos theorem

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Dylanger S. Pittman, “A constructive proof of the Bollobás–Varopoulos theorem,” arXiv:2110.11336, version 2 (stamped 18 December 2021). The first page's footer reads "Preprint submitted to Journal of Mathematical Analysis and Applications" beside the date December 21, 2021 (p. 1); no journal publication is asserted here.

The measure criterion

Theorem 1.1 (PDF p. 1) considers a finite non-atomic measure $(\Omega,\mathcal S,\nu)$, a measurable set $A$ with $\nu(A)>0$, measurable subsets $A_1,\ldots,A_n\subseteq A$, and positive target masses $m_1,\ldots,m_n$. There are pairwise disjoint measurable sets $B_k\subseteq A_k$ with $\nu(B_k)=m_k$ for every $k\in[n]$ if and only if

$$\nu\left(\bigcup_{i\in I}A_i\right)\geq\sum_{i\in I}m_i \quad\text{for every }I\subseteq[n].$$

Corollary 2.3 (PDF p. 3), which the paper identifies as Exercise 2.9 on p. 54 of Diestel's Graph Theory and states without proof as an application of Hall's matching theorem, says that for a finite set $A$ with subsets $A_i$ and $d_i\in\mathbb N$, pairwise disjoint $D_k\subseteq A_k$ with $|D_k|=d_k$ exist exactly when

$$\left|\bigcup_{i\in I}A_i\right|\geq\sum_{i\in I}d_i \quad\text{for every }I\subseteq[n].$$

Corollary 2.4 on the same page is the equal-weight restatement. With $\xi>0$ and the discrete measure $\eta(X)=\xi|X|$ on $2^A$, it asks for disjoint $D_k\subseteq A_k$ with $\eta(D_k)=\xi d_k$ under the equivalent inequalities $\eta(\bigcup_{i\in I}A_i)\geq\xi\sum_{i\in I}d_i$.

The finite weighted form, Corollary 2.5 (PDF p. 4), applies the same Hall criterion to a finite collection of disjoint measurable pieces of common measure $\xi$. For admissible families of those pieces, disjoint selections of total mass $\xi d_k$ exist exactly when every subfamily union has mass at least $\xi$ times the corresponding sum of demands.

Constructive route and scope

The proof of Theorem 1.1 begins by partitioning the Boolean atoms $S_Q$, $\varnothing\neq Q\subseteq[n]$, into small equal-measure pieces. The print (p. 5) defines $S_Q=(\bigcap_{i\in Q}A_i)\setminus(\bigcap_{i\notin Q}A_i)$; the disjointness of distinct $S_Q$ and the partition identities stated next hold for the Venn atoms

$$S_Q=\left(\bigcap_{i\in Q}A_i\right)\setminus \left(\bigcup_{i\notin Q}A_i\right),$$

so the second intersection is read as a union. The proof applies the finite weighted Hall form (Corollary 2.5) to the resulting finite families, passes through a nested sequence as the mesh tends to zero, and removes null overlaps in the limiting sets. This is a route summary rather than a complete proof transcription; the proof occupies pp. 5–8.

For explicit compilation-level method context only, see Ford–Fulkerson's representative-system flow source. Pittman does not cite that source, and the measurable splitting construction should remain a separate method. The paper supports no numbered Erdős problem connection, and this selected statement digest carries no complete-proof credit.

The retained source is the arXiv v2 PDF. The arXiv record (https://arxiv.org/abs/2110.11336, read 2026-10-02) names the Creative Commons Attribution 4.0 license.