Betti Bounds and Nerve HVC Dimension

Definitions and mathematical statement

Let $\mathcal M$ be a family of subsets of ${\mathbb{R}}^d$. Its nerve is the simplicial complex \[ N(\mathcal M)=\{\mathcal J\subseteq\mathcal M: \mathcal J\text{ finite and }\bigcap_{M\in\mathcal J}M\ne\varnothing\}, \] with the empty face included by convention. The source asks whether, for fixed $A>0$, the uniform intersection bound \[ b(L_{\mathcal J})\leq A\binom{m}{d},\qquad L_{\mathcal J}=\bigcap_{M\in\mathcal J}M,\quad|\mathcal J|=m, \] forces the nerve to have HVC dimension $d$. The constant must be fixed across all intersections and relevant values of $m$, rather than chosen separately for each intersection. The record does not specify finiteness of the family, the $m$ range or regularity assumptions on its subsets.

The symbols $b$ and HVC dimension are not defined in the supplied text. If $b$ means a Betti-number sum, the coefficient field and reduced versus unreduced convention must be fixed. For example, an unreduced total Betti number of a nonempty space includes its nonzero $H_0$, whereas a reduced sum need not. The definition of HVC dimension, including whether "dimension $d$" means equality or an upper bound, is essential; it is not replaced here by ordinary VC dimension, Helly number or Leray dimension.

There is a material small-$m$ issue. With the usual binomial convention, $\binom md=0$ for $m<d$. If the hypothesis applies to those $m$ and $b$ is unreduced total Betti number, it forces every such intersection empty. In particular, for $d\geq2$, applying it to one member excludes every nonempty member. The intended convention may instead involve reduced Betti numbers, a restricted $m$ range or different surrounding notation, but none is silently chosen as a correction. The treatment of empty intersections also needs specification.

The stated conjectural implication is preserved with these unresolved definitions and quantifiers. The literature triage checked 21 August 2026 records the nerve HVC assertion as open and says no general theorem for all families under the stated bound was found. It supplies no definition resolving these issues or exact theorem to be proved. The draft therefore does not certify a well-posed all-subsets implication merely from the corrected-verified metadata.

Short English statement

The source proposes a nerve-dimension consequence of uniform Betti bounds, but does not define HVC dimension or the Betti convention. The range $m<d$ is especially significant because the printed binomial bound vanishes there. Those choices must be retained as unresolved rather than silently repaired.

Sources

• Local UnsolvedMath registry, permanent ID 30000084, source code catalogue entry, supplied Oberwolfach Reports record; updated 16 September 2026.
• Supplied citation: Combinatorial Commutative Algebra (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/32, pp. 1703--1746: https://doi.org/10.4171/owr/2004/32. Proposed year is separately 2004.
• Kalai, Combinatorial Expectations, supplied reference: https://gilkalai.wordpress.com/wp-content/uploads/2019/03/ow04.pdf.
• Literature triage checked 21 August 2026, derived from the supplied OWR curation; independently verify time-sensitive status and cited claims. No full invariant definition or theorem statement is supplied.