Graded Radon and Helly Numbers

• Graded Radon and Helly numbers are generalizations of classical convexity invariants that measure local intersection complexities in set systems under topological and algebraic constraints.
• They refine intersection analysis by evaluating graded parameters over subfamilies, leading to sharper fractional Helly-type results and resolution of longstanding conjectures.
• Applications include mixed integer spaces, polynomial systems, and topological combinatorics, offering new insights into classical intersection problems.

Graded Radon and Helly numbers are generalizations of classical convexity invariants designed to capture the combinatorial geometry of set systems and convexity spaces, especially under nontrivial topological or algebraic constraints. They enable fine-grained analysis of intersection phenomena and threshold behaviors in families of sets, providing hierarchical or “graded” alternatives to conventional Radon, Helly, and fractional Helly numbers. This graded perspective has led to sharper theorems verifying fractional Helly-type results and underpinning long-standing conjectures even when the original (ungraded) numbers are unbounded, particularly in regimes where intersection patterns are controlled in aggregate over small subfamilies.

1. Classical Background and the Notion of Grading

Classical Radon and Helly numbers quantify intersection properties in convexity spaces:

• The Radon number $r$ is the minimum $m$ such that any set of $m$ points can be partitioned into two parts whose convex hulls intersect.
• The Helly number $h$ is the smallest $h$ such that for any collection of convex sets, if every $h$ sets have nonempty intersection, then the full family does.

These parameters have well-known bounds in spaces like $\mathbb{R}^d$ and $\mathbb{Z}^d$, e.g., $h(\mathbb{R}^n) = n+1$, $h(\mathbb{Z}^d)=2^d$ (Averkov et al., 2010). Fractional variants (fractional Helly numbers) relax local intersection requirements, guaranteeing a large intersecting subfamily under density assumptions (Patáková, 2019).

Graded Radon and Helly numbers extend these invariants by considering their growth across all subfamilies of bounded size, rather than the global parameter alone. For a set system $F$, the graded Radon number at level $t$ is

$$\text{rad}_F(t) = \sup\{ \text{rad}_G \mid G \subset F,\, |G|\le t \},$$

and similarly for $\text{h}_F(t)$.

This “grading” approach provides a spectrum of complexity measures, making it possible to detect and exploit intersection phenomena even when global obstructions exist or global numbers diverge (Bin, 2024, Avvakumov et al., 6 Jan 2026).

2. Formal Definitions: Graded Parameters, Shatter Functions, and Betti Complexity

Let $F$ be a (possibly infinite) family of subsets over a ground set $X$. The graded invariants are defined by maximizing classical parameters over all subfamilies of at most $t$ elements:

• Graded Radon number: $$\text{rad}_F(t) = \sup\{ \text{rad}_G : G \subset F, |G| \le t \}$$.
• Graded Helly number: $$\text{h}_F(t) = \sup\{ \text{h}_G : G \subset F, |G| \le t \}$$.
• Graded colorful Helly number: analogous, maximizing the colorful Helly number over all size-$t$ subfamilies (Bin, 2024, Avvakumov et al., 6 Jan 2026).

For set systems with topological constraints, the homological shatter function $\phi_F^{(h)}(t)$ measures the aggregate Betti complexity of intersections among all size-$t$ subfamilies:

$$\phi_F^{(h)}(t) = \sup \left\{ \sum_{i=0}^h \beta_i(\cap G) \mid G \subset F, |G|=t \right\},$$

with all Betti numbers taken over $\mathbb{Z}_2$ (Bin, 2024).

These graded numbers reflect “local” to “global” behaviors. When the growth of graded Radon or Helly functions is sublinear or logarithmic, finite global parameters are often forced (Avvakumov et al., 6 Jan 2026).

3. Core Theorems and Growth Phenomena

Graded parameters satisfy sharp inequalities analogous to classical results:

• Graded Levi inequality: $\text{h}_F(t) \leq \text{rad}_F(t) - 1$ for every $t$ (Avvakumov et al., 6 Jan 2026, Bin, 2024).
• Sublinear Helly growth: If there exists $t_0$ such that $\text{h}_F(t)<t$ for all $t>t_0$, then $\text{h}_F \leq t_0$ (i.e. the global Helly number becomes finite) (Avvakumov et al., 6 Jan 2026).
• Discrete jump bound: If $\text{rad}_F(t) > \text{rad}_F(t-1)$, then $$\text{rad}_F(t-1) \geq 1 + \log_2 \left(1 + \frac{t}{\text{h}_F(t)} \right),$$ exhibiting a minimum jump rate as graded Radon increases (Avvakumov et al., 6 Jan 2026).
• Ultimate slow growth: If $$\lim_{t \to \infty} (\text{rad}_F(t) - \log_2 t) = -\infty$$, then the ordinary Radon number is finite (Avvakumov et al., 6 Jan 2026).

For set systems with bounded or slowly growing homological shatter functions, one obtains:

• Bounded Radon and Helly numbers as functions of the Betti-bound $b$ and the ambient dimension $d$ (Patáková, 2019).
• Fractional Helly theorems for families where the shatter function grows slower than certain threshold functions $V_{d,b}(t)$, which interpolate between iterated logarithm and inverse Ackermann rates (explicit in (Bin, 2024)).
• Verification of fractional Helly theorems and $(p,q)$-theorems for systems with controlled graded parameters or Betti complexity, even when global numbers diverge (Bin, 2024, Avvakumov et al., 6 Jan 2026).

4. Connections to Fractional Helly, Tverberg, and Topological Constraints

Linear and sublogarithmic growth of graded Radon numbers directly imply fractional Helly-type results (Bin, 2024, Avvakumov et al., 6 Jan 2026). By leveraging graded bounds, it is possible to:

• Prove the existence of large intersecting subfamilies when almost all small subfamilies intersect, generalizing Matoušek’s shatter function theorem to the homological/topological regime.
• Recover Tverberg-type and colorful Helly-type theorems with parameters governed by the graded Radon and Helly numbers, rather than the much larger global ones (Pálvölgyi, 2019, Bin, 2024).
• Connect to weak $\epsilon$-nets and $(p,q)$-piercing thresholds via the fractional Helly/Levi machinery (Patáková, 2019, Bin, 2024).

The precise threshold for global finiteness (e.g., Radon number becoming finite) is sublogarithmic growth in the graded function. Example constructions demonstrate families with graded Radon numbers growing as $\Theta(\log t)$, yet their global Radon number may still be finite given further constraints (Bin, 2024).

For algebraic systems (ideals, polynomial varieties), graded Helly numbers are governed by combinatorial dimension, with the bound $H(n,d) = \binom{n+d}{d}$ in degree-$d$ polynomials in $n$ variables (Loera et al., 2015).

5. Proof Strategies and Methodological Insights

Proofs exploiting graded parameters commonly combine:

• Probabilistic and hypergraph partitioning techniques to transfer local intersection density into global structure (Pálvölgyi, 2019).
• Topological chain map arguments leveraging bounded Betti numbers, especially via van Kampen–Flores analogues, to enforce Radon-type partitioning in high dimensions (Patáková, 2019, Avvakumov et al., 6 Jan 2026).
• Ramsey-theoretic and supersaturation arguments to upgrade intersecting substructures (e.g., $k$-wise intersection) into clique or partition structures (Patáková, 2019, Pálvölgyi, 2019).

The proof of the main linear bound for the $k$-th Radon number in a convexity space,

$r_k \leq c(r_2) k$

combines Bukh’s hypergraph matching with the abstract fractional Helly theorem of Holmsen and Lee to show that increased intersection density forces the existence of partitioned subfamilies with common intersection, with $c(r_2)$ admitting explicit (but large) dependency on the Radon number $r_2$ (Pálvölgyi, 2019).

For graded fractional Helly, a multi-step reduction is used: control the graded Helly and colorful Helly numbers via the shatter function, apply combinatorial fractional Helly lemmas (Holmsen, Lee) on induced subfamilies, and thus extract large intersecting subfamilies when local intersection density is positive (Bin, 2024, Avvakumov et al., 6 Jan 2026).

6. Applications, Examples, and Ongoing Questions

Applications span computational algebra, convexity theory, and topological combinatorics:

• In mixed integer spaces ($\mathbb{R}^n \times \mathbb{Z}^d$), both Helly and Radon numbers have explicit bounds: $h = (n+1) 2^d$, $r$ between $(n+1)2^d + 1$ and $(n+d)(n+1)2^d - n - d + 2$ (Averkov et al., 2010).
• For polynomial systems, the graded Helly number coincides with the system’s combinatorial rank; in plane curves of degree $\leq d$, Helly number is $\binom{2+d}{2}$ (Loera et al., 2015).
• In topological settings, bounding homological shatter functions forces finiteness of graded and global Radon/Helly numbers, enabling the verification of the Kalai–Meshulam conjecture in new regimes (Patáková, 2019, Bin, 2024, Avvakumov et al., 6 Jan 2026).

Open questions include optimizing the constant factors in linear Radon growth for abstract convexity spaces, tightening thresholds in graded fractional Helly, and refining algebraic analogues for graded Radon-type statements. The interplay between homological/topological restrictions and combinatorial growth rates remains an active area, with potential for new fractional Helly-type theorems sensitive to graded intersection complexity (Bin, 2024, Avvakumov et al., 6 Jan 2026).

7. Summary Table: Graded Invariants and Their Properties

Parameter	Classical Definition	Graded Definition over subfamilies of size $t$	Typical bound in $\mathbb{R}^d$
Radon number $r$	Min $m$ for Radon partition	$\text{rad}_F(t) = \sup_{|G|\le t} \text{rad}_G$	$d+2$
Helly number $h$	Min $h$ for Helly property	$\text{h}_F(t) = \sup_{|G|\le t} \text{h}_G$	$d+1$
Fractional Helly number	Min $h$ for density-based intersections	Analogous, maximized over size-$t$ subfamilies	$d+1$
Homological shatter	Max Betti sum for $k$-wise intersections	$$\phi_F^{(h)}(t) = \sup_{|G|=t} \sum_{i=0}^h \beta_i(\cap G)$$	$=0$ for convex sets

Slow growth of $\text{rad}_F(t)$ or $\phi_F^{(d)}(t)$ forces bounded global parameters and validates fractional Helly results in various geometric, topological, and algebraic contexts (Bin, 2024, Avvakumov et al., 6 Jan 2026, Loera et al., 2015, Pálvölgyi, 2019).