Fractional Helly Theorem and Extensions

• Fractional Helly theorem is a key geometric result linking frequent local intersections in convex sets to a large globally intersecting subfamily.
• The colorful fractional Helly theorem partitions sets into color classes, providing optimized intersection bounds through combinatorial and hypergraph methods.
• Recent advances using weak saturation and algebraic techniques establish sharp thresholds, enhancing applications in computational geometry and combinatorial theory.

The fractional Helly theorem is a fundamental result in discrete and computational geometry describing the relationship between frequent local intersections among families of convex sets and the existence of a substantial globally intersecting subfamily. The classical theorem and its generalizations, such as the colorful fractional Helly theorem, are central in the theory of combinatorial convexity, with deep connections to extremal combinatorics, topology, and related Helly-type theorems. The colorful fractional Helly theorem merges two classical generalizations—the colorful Helly theorem and the fractional Helly theorem—yielding sharpened insights into intersection patterns when families are partitioned into color classes.

1. Classical and Colorful Fractional Helly Theorems

The classical Helly theorem states that for a finite family of convex sets in $\mathbb{R}^d$, if every %%%%1%%%%-tuple has nonempty intersection, then the whole family has nonempty intersection. The fractional Helly theorem, originally established by Katchalski and Liu, relaxes "every" to "a positive fraction":

Classical Fractional Helly Theorem (Katchalski–Liu, Kalai, Eckhoff):

Let $d \geq 1$ and $\alpha \in (0,1]$. There exists $\beta=\beta(\alpha,d) \in (0,1]$ such that for any finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$ with $n=|\mathcal{F}| \geq d+1$, if at least $\alpha \binom{n}{d+1}$ of the $(d+1)$-tuples of $\mathcal{F}$ intersect, then $\mathcal{F}$ contains an intersecting subfamily of size at least $\beta n$. The optimal value is: $\beta(\alpha,d) = 1 - (1 - \alpha)^{1/(d+1)}$ which is sharp (Kim, 2015).

The colorful Helly theorem, due to Lovász, considers $d+1$ families ("color classes") and asserts that if every rainbow (colorful) $(d+1)$-tuple has nonempty intersection, then one whole color class is intersecting.

The colorful fractional Helly theorem, as developed by Bárány, Fodor, Montejano, Oliveros, and Pór, extends this to a fractional scenario:

Colorful Fractional Helly Theorem (Bárány et al., 2014):

Let $d \geq 1$, $\alpha \in (0,1]$, and let $\mathcal{F}_1, \dots, \mathcal{F}_{d+1}$ be finite nonempty families of convex sets in $\mathbb{R}^d$. If at least $\alpha \prod_{i=1}^{d+1} |\mathcal{F}_i|$ of the colorful $(d+1)$-tuples have nonempty intersection, then for some $i$, the family $\mathcal{F}_i$ contains an intersecting subfamily of size at least $(\alpha/(d+1)) |\mathcal{F}_i|$, i.e.,

$$\beta_{\mathrm{col}}(\alpha, d) = \frac{\alpha}{d+1}$$

This bound remains strictly less than $1$ even as $\alpha \to 1$, failing to recover the full colorful Helly theorem (Kim, 2015).

2. Sharpened Bounds and Optimality

Minki Kim (Kim, 2015), and independently Bulavka, Goodarzi, and Tancer (Bulavka et al., 2020), demonstrated that the optimal dependence in the colorful fractional Helly theorem matches that of the non-colorful fractional Helly theorem: $$\beta_{\mathrm{col}}(\alpha, d) = 1 - (1-\alpha)^{1/(d+1)}$$ This implies $\beta_{\mathrm{col}}(\alpha, d) \to 1$ as $\alpha \to 1$, restoring the full strength of the colorful Helly theorem in the limit and providing optimal constants for all $\alpha$ (Bulavka et al., 2020).

Proof Sketch (Extending Kalai's Exterior Algebra Method)

• The proof reformulates the intersection problem via nerves and colored simplicial complexes: each color class forms a part in the vertex set of the complex, where a face represents a subfamily with nonempty intersection.
• The nerve complex of convex families is $d$-collapsible.
• By counting the number of colorful subfaces using algebraic and combinatorial methods, and bounding the number of non-intersecting subfamilies, the sharp threshold is established (Bulavka et al., 2020).

Tightness: For each $i$, let $\mathcal{F}_i$ contain $(1-\gamma) n_i$ copies of $\mathbb{R}^d$ and $\gamma n_i$ generic hyperplanes such that $\gamma = (1-\alpha)^{1/(d+1)}$. Any colorful $(d+1)$-tuple with all hyperplanes is non-intersecting, and the largest intersecting subfamily in any color class has size $(1-\gamma)n_i$, giving tightness (Bulavka et al., 2020).

3. Weak Saturation, Extremal Combinatorics, and Short Proofs

Recent developments by Chakraborti, Cho, Kim, and Kim connect the optimal bounds for both the classical and colorful fractional Helly theorems to weak saturation in extremal hypergraph theory (Chakraborti et al., 2024). In this framework:

• The nerve of a finite family of convex sets in $\mathbb{R}^d$ is $d$-collapsible.
• The fractional Helly property mirrors the minimal weak saturation number for certain multipartite hypergraphs.
• These reductions recover the optimal bounds, utilizing only linear algebraic arguments and avoiding exterior-algebraic machinery (Chakraborti et al., 2024).

The optimal enumerative form is also obtained: for $n_i$ sets in each class and size parameters $r_i \leq n_i$,

$$\#\{\text{intersecting rainbow } (d+1)\text{-tuples}\} \leq \prod_{i=1}^{d+1} n_i - \prod_{i=1}^{d+1} (n_i - r_i)$$

Exceeding this bound implies the existence of an intersecting subfamily of size $r_i+1$ in some color class (Chakraborti et al., 2024).

4. Extensions: Mixed-Colorful Case, Cartesian Products, and Non-Euclidean Settings

The optimal threshold generalizes to the "mixed colorful" case: for $c$ colors, selection vector $k=(k_1, ..., k_c)$, and thresholds $\beta_i$, the sharp condition reads

$$\alpha_k(d, k, \beta) = \sum_{\ell \in L(d)} \prod_{i=1}^c \binom{k_i}{\ell_i} (1-\beta_i)^{\ell_i} (\beta_i)^{k_i - \ell_i}$$

where $$L(d) = \{\ell \in \mathbb{N}^c : \sum_{i=1}^c \ell_i \leq d, 0 \leq \ell_i \leq k_i\}$$ (Bulavka et al., 2020).

In product settings, the theorem applies to families of Cartesian products of convex sets in $\mathbb{R}^{td}$ (Chakraborti et al., 2021): If at least an $\alpha$-fraction of the $(d+1)$-tuples of such products intersect (for $\alpha > 1 - 1/t^d$), then at least a $(1 - (t^d(1-\alpha))^{1/(d+1)})$-fraction is intersecting. This is sharp, and further intersection conditions do not yield stronger bounds (Chakraborti et al., 2021).

The colorful fractional Helly theorem has also been extended to Banach spaces with uniformly convex norms, with the optimal dependence preserved up to constants determined by the modulus of convexity (Ivanov, 2024).

5. Impact, Generalizations, and Applications

The determination of sharp quantitative and qualitative constants in the fractional Helly and colorful fractional Helly theorems enables refined analyses of transversal numbers, $(p, q)$-theorems, and the construction of weak $\varepsilon$-nets in both geometric and abstract convexity spaces. The optimal bounds are essential for tight guarantees in applications including centerpoint theorems, computational algorithms for high-dimensional geometric data, and combinatorial incidence problems.

The sharp enumeration and saturation principles provide new tools for bridging geometry and extremal combinatorics, and the reductions to weak saturation open the way to further short proofs and structural insights in Helly-type theory (Chakraborti et al., 2024).

6. Comparison Table: Classical vs. Colorful Fractional Helly Bounds

Theorem Variant	Condition on $\alpha$-fraction	Minimum Intersecting Subfamily Size	Behavior as $\alpha \to 1$
Classical Fractional Helly	$(d+1)$-tuples	$1 - (1-\alpha)^{1/(d+1)}$	Tends to $1$
Colorful Fractional Helly (Bárány et al.)	Colorful $(d+1)$-tuples	$\alpha/(d+1)$	Tends to $1/(d+1)$
Colorful Fractional Helly (Optimal)	Colorful $(d+1)$-tuples	$1 - (1-\alpha)^{1/(d+1)}$	Tends to $1$

The use of weak saturation, exterior algebra, and combinatorial hypergraph methods are central to recent advances, with exact thresholds realized by extremal constructions (Kim, 2015, Bulavka et al., 2020, Chakraborti et al., 2024).

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References:

• "A note on the colorful fractional Helly theorem" (Kim, 2015)
• "Optimal bounds for the colorful fractional Helly theorem" (Bulavka et al., 2020)
• "Colorful fractional Helly theorem via weak saturation" (Chakraborti et al., 2024)
• "Fractional Helly theorem for Cartesian products of convex sets" (Chakraborti et al., 2021)
• "No-dimensional Helly's theorem in uniformly convex Banach spaces" (Ivanov, 2024)