Minimum Venn Diagrams: Bounds & Constructions

• Minimum Venn diagrams are arrangements of n simple closed curves in the plane that ensure every one of the 2^n intersections is represented by a single connected region with the minimum crossing count L_n.
• These diagrams are constructed using combinatorial techniques such as dual hypercube encoding and isometric cycle partitions to closely approach the theoretical lower bound.
• Recent advances provide explicit constructions for n ≤ 7 and near-optimal designs for larger n, while challenges remain in refining Gray code strategies and proving the Bultena–Ruskey conjecture.

A minimum Venn diagram is an arrangement of $n$ simple closed curves in the plane meeting finitely many times so that each of the $2^n$ possible intersections is represented by a single connected region, and the set of crossings is as small as possible. The minimum number of crossings in an $n$-Venn diagram is denoted $L_n := \lceil (2^n-2)/(n-1) \rceil$. The classical case—where every crossing is between just two curves—achieves the maximal number of crossings, $2^n-2$. The minimum case, where each crossing involves all $n$ curves, is considerably more subtle, and such diagrams (if they exist) are called minimum Venn diagrams. These are known only for $n \leq 7$, though asymptotically sharp bounds and near-minimal explicit constructions are now available for all $n \geq 8$ (Brenner et al., 12 Nov 2025).

1. Structural Constraints and the Bultena–Ruskey Bound

For any $n \geq 2$, Bultena and Ruskey established a lower bound on the total number of crossings achievable in an $n$-Venn diagram:

$L_n = \Bigl\lceil \frac{2^n-2}{n-1} \Bigr\rceil$

Their argument relies on the observation that while introducing a crossing between $d$ curves can at most generate $d$ new connected regions, the aim is to create $2^n$ regions, requiring at least $(2^n-1)/(n-1)$ crossings. A parity correction leads to the $-2$ term and the ceiling operation.

Attaining the lower bound necessitates that every crossing involve all $n$ curves, maximizing the “productivity” of each intersection. Diagrams realizing $L_n$ crossings are called minimum Venn diagrams. Such diagrams are explicitly constructed for $n \leq 7$, and their existence for all $n \geq 8$ remains the subject of the Bultena–Ruskey conjecture (Brenner et al., 12 Nov 2025).

2. Explicit Constructions Near the Minimum

For $n=8$, the theoretical lower bound is $L_8=37$, yet the best previously known construction achieved $42$ crossings. Using a combinatorial construction based on dual hypercube structures and careful cycle composition, an explicit $8$-Venn diagram with $40$ crossings has been constructed, which is within $3$ of the minimal possible (Brenner et al., 12 Nov 2025).

Further, for $n=2^k$ with $k \geq 4$, there exist explicit constructions with

$(1+\frac{33}{8n}) L_n = (1+o(1))L_n$

crossings, so the ratio of actual to minimum crossings tends to $1$ as $n \to \infty$.

This represents an asymptotic completion of the Bultena–Ruskey conjecture: for all sufficiently large $n$, minimum Venn diagrams with $(1+o(1))L_n$ crossings can always be constructed, though the exact bound is still unattained for all individual $n$.

3. Methodological Framework: Hypercube Duality and Isometric Cycle Partitions

Central to these constructions is an encoding of $n$-Venn diagrams as planar, spanning, degree-$n$ subgraphs of the $n$-dimensional hypercube $Q_n$. Here, each region of the Venn diagram can be identified with a vertex in $Q_n$ (a subset of $[n]$), and the adjacency of regions corresponds to edges of $Q_n$. Each crossing in the diagram corresponds to a face in the dual planar subgraph, typically a $2\ell$-cycle with $\ell \in \{2, \dots, n\}$.

When $n$ is a power of two, Ramras’s 1992 result enables partitioning $Q_n$ into $\frac{2^n}{2n}$ disjoint isometric $2n$-cycles. By arranging these cycles concentrically in the plane and connecting them via specifically designed gadgets, and then merging 6-faces into larger faces by careful edge deletions along “long-run” Gray codes, one achieves a planar graph with a number of faces closely tracking $L_n$.

An explicit basis $C_k$ of the relevant linear subspace of $\mathbb{Z}_2^n$ is used to maximize the contiguous runs in the flip sequence of the Gray code, controlling the deficiency from the lower bound. The number and total length of maximal runs directly determine the additive error above $L_n$ in the crossing count (Brenner et al., 12 Nov 2025).

4. Generalizations via the Doubling Trick

Given an $n$-Venn diagram whose dual outer face is “colorful” (contains two antipodal vertices of $Q_n$), it is possible to add an $(n+1)$\textsuperscript{th} curve in such a way that the number of crossings precisely doubles, while maintaining the required properties. Repeated application of this operation allows the construction of $(n+m)$-Venn diagrams with

$40 \cdot 2^m \quad\text{crossings for }n=8$

and for all $n=2^k$ with $k\geq 4$,

$$(1+\frac{33}{8n})\frac{n+m}{n}L_{n+m} = (2+o(1))L_{n+m}$$

crossings for all $0 \leq m < n$. Thus, explicit near-minimum diagrams exist for all $N \geq 8$, and the total crossings are at most $2L_N$ asymptotically.

5. Extremal Set Families and Dual VC-dimension

From a broader combinatorial perspective, Venn diagrams are closely linked to extremal questions concerning set systems and dual VC-dimension. For families $\mathcal{F} \subseteq \{0,1\}^n$, the dual VC-dimension, $\mathrm{VC}_{\text{dual}}(\mathcal{F})$, quantifies the largest $k$ for which there exist $k$ sets whose Venn diagram (the $2^k$ regions determined by their intersections) is full, i.e., all regions are non-empty.

The extremal result states that for some absolute constant $C>0$, if $|\mathcal{F}| \geq Cn^3$, then $\mathrm{VC}_{\text{dual}}(\mathcal{F}) \geq 3$, or equivalently, $\mathcal{F}$ contains three sets whose 3-set Venn diagram is full. The exponent $3$ is shown to be optimal up to the constant, as taking all subsets of size $\leq 2$ (yielding $\Theta(n^2)$) avoids a full 3-Venn diagram (Keevash et al., 2019).

This connects to the dual Sauer–Shelah lemma: the size threshold $n^3$ demarcates the transition from dual VC-dimension $\leq 2$ to $\geq 3$, echoing structural rigidity at the level of the incidence geometry of set families.

6. Open Problems and Asymptotic Gaps

Several major problems remain open:

• It is unknown whether minimum Venn diagrams with exactly $L_n$ crossings exist for every $n \geq 8$ (the Bultena–Ruskey conjecture).
• For $n=8$, the current best is $40$ crossings, three above $L_8=37$.
• Improving or eliminating the error constant $33/8$ is contingent on finding Gray codes with even longer contiguous runs within the cube coefficient space, which would immediately refine the construction.
• The enumeration of non-simple or minimum diagrams for small $n$ is unsettled.
• The question of whether every simple Venn diagram extends to a higher-order diagram is known to have counterexamples for $n \geq 6$.
• In the combinatorial direction, determining the exact order of growth $f_k(n)$, the largest size of a set family with dual VC-dimension at most $k$, remains open for $k \geq 3$. The conjecture suggests $f_k(n) = \Theta(n^{2^{k-1}-1})$ (Keevash et al., 2019, Brenner et al., 12 Nov 2025).

7. Quantitative Summary for Small $n$

A representative table for $N=8$ through $16$ shows the quantitative performance of the best constructions:

$N$	$L_N$	Best known (Brenner et al., 12 Nov 2025)	Monotone-diagram bound $\binom{N}{\lfloor N/2 \rfloor}$
$8$	$37$	$40$	$70$
$9$	$64$	$80$	$126$
$10$	$114$	$160$	$252$
$11$	$205$	$320$	$462$
$12$	$373$	$640$	$924$
$13$	$683$	$1280$	$1716$
$14$	$1261$	$2560$	$3432$
$15$	$2341$	$5120$	$6435$
$16$	$4369$	$5118$	$12870$

For each $N$, $L_N$ is the Bultena–Ruskey minimum, “Best known” is the construction provided in (Brenner et al., 12 Nov 2025), and for comparison the monotone-diagram bound is given by $\binom{N}{\lfloor N/2 \rfloor}$.

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