Fractional Helly Theorems for Bounded HVC Dimension

Definitions and mathematical statement

For a finite simplicial complex $K$ on $n$ vertices, let $f_d(K)$ count faces of cardinality $d+1$ and let $\dim K=\max_{\sigma\in K}(|\sigma|-1)$. Betti numbers are understood relative to one fixed coefficient convention, for example $b_i(K)=\dim_{\mathbb F}H_i(K;\mathbb F)$ over a fixed field $\mathbb F$.

The verified source formulation fixes a strongly hereditary class $\mathcal K$, a constant $A>0$ and a dimension parameter $d$, and assumes that every $n$-vertex $K\in\mathcal K$ satisfies \[ \sum_i b_i(K)<An^d. \] It asks whether this implies the following fractional Helly assertion, in the source's order-$d$ terminology: \[ \forall t>0\ \exists s(t)>0\ \forall K\in\mathcal K, \qquad f_d(K)\ge t\binom n{d+1} \ \Longrightarrow\ \dim K\ge s(t)n, \quad n=|V(K)|. \] The constant $s(t)$ may depend on the fixed class, $A$ and $d$, but must be independent of $n$ and the particular $K$. Thus a positive fraction of possible $d$-faces should force a face whose dimension is a positive linear fraction of the vertex count. The original short statement calls the hypothesis homological VC dimension $d$ with constant $A$.

The extract does not define "strongly hereditary", specify its extra closure operations, choose ordinary versus reduced Betti numbers, or fix coefficients. These conventions must be recovered from the source rather than assuming closure under a particular unlisted operation. There is also a finite-size qualification to check: if $n<d+1$, then $\binom n{d+1}=0$, so the printed antecedent can hold vacuously without a $d$-face. The source must specify a nonvacuous range, sufficiently large $n$, or an appropriate small-size adjustment. The literal conclusion uses $\dim K$, rather than largest-face cardinality $\dim K+1$; this distinction is retained instead of silently changing the inequality.

The literature assessment checked on 21 August 2026 reports the bounded-homological-VC fractional Helly conjecture as a research question and says known $d$-Leray results cover only a special class. Here $d$-Leray conventionally means that every induced subcomplex has vanishing reduced homology in degrees $i\ge d$ over fixed coefficients. The source's more general Betti-growth hypothesis is not replaced by that stronger special-case assumption. The open label is dated triage requiring independent verification.

Short English statement

Does a uniform polynomial bound on total Betti numbers in a strongly hereditary class force a fractional Helly theorem? A positive density of $d$-faces should produce a face of linear size, with constants uniform over the class; the closure, homology and small-size conventions need clarification.

Sources

• 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 supplied as 2004.
• Literature assessment checked 21 August 2026: Kalai, Combinatorial Expectations, https://gilkalai.wordpress.com/wp-content/uploads/2019/03/ow04.pdf; "Original Oberwolfach report," https://doi.org/10.4171/owr/2004/32.
• Local UnsolvedMath registry, permanent ID $30000083$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. The quantitative clean statement is labelled corrected and verified against the report.