Fractional Expectation Threshold in Random Structures

• Fractional expectation threshold is a measure quantifying the tightest fractional lower bound for threshold phenomena in monotone set systems using optimal linear programming relaxations.
• It leverages LP duality and selector processes to bridge the gap between true probabilistic thresholds and expectation thresholds, ensuring sharp and practical estimates.
• Applications include analyzing random hypergraph matchings, bounded degree spanning trees, and resolving conjectures such as those of Kahn–Kalai and Talagrand.

The fractional expectation threshold is a fundamental concept in probabilistic combinatorics, quantifying the tightest “fractional” first-moment lower bound for threshold phenomena in random discrete structures, especially those defined by monotone set systems. It appears as the optimal value of a fractional covering linear program and is intrinsically related to both the true probabilistic threshold and the so-called expectation threshold, which is derived from integral (i.e., set-system) coverings. The interplay between the fractional and expectation thresholds lies at the heart of threshold sharpness, universality, and the resolution of major conjectures such as those of Kahn–Kalai and Talagrand.

1. Formal Definitions and Linear Programming Formulation

For an $n$-element ground set $X$ and an increasing (monotone) family $F \subseteq 2^X$, the fractional expectation threshold quantifies, for a given $p \in [0,1]$, the “fractional covering” of $F$ using sets, each “bought” at cost $p^{|S|}$. Let $g:2^X \rightarrow [0,1]$ assign a weight to each subset. The fractional threshold is

$$T_{\rm frac}(F) := \sup\left\{ p : \exists\, g:2^X\to[0,1],\; F \subseteq \langle g \rangle,\, w(g,p) \le \tfrac12 \right\}$$

where

$$w(g,p) := \sum_{S \subseteq X} g(S)\,p^{|S|}, \quad \langle g \rangle := \left\{ S \subseteq X : \sum_{T \subseteq S} g(T) \ge 1 \right\}$$

The LP relaxation allows coverings to be fractional while an analogous integral version defines the expectation threshold $T_{\rm exp}(F)$, minimizing over set-systems $G \subseteq 2^X$:

$$T_{\rm exp}(F) := \sup\left\{ p : \exists\, G \subseteq 2^X,\; F \subseteq \langle G \rangle,\, w(G,p) \le \tfrac12 \right\}$$

with

$w(G,p) := \sum_{S \in G} p^{|S|}$

By definition, $T_{\rm exp}(F) \le T_{\rm frac}(F)$ always (Fischer et al., 21 Oct 2025, Frankston et al., 2019, Pham, 4 Dec 2024).

2. Threshold Phenomena and Major Theorems

In most classical problems, one studies a monotone property $\mathcal{F} \subseteq 2^X$ under the product measure $\mu_p$, where each $x \in X$ is independently included with probability $p$. The probabilistic threshold $p_c(\mathcal{F})$ is defined by $\mu_{p_c}(\mathcal{F}) = 1/2$.

A series of major breakthroughs have established near-optimal relationships:

• Fractional Threshold Theorem: For the minimal size $\ell = \max_{A \in \mathcal{F}_{\min}} |A|$,

$$p_c(\mathcal{F}) \le K\, T_{\rm frac}(\mathcal{F})\, \log \ell$$

for a universal constant $K$ (Frankston et al., 2019). This framework leverages LP duality, fractional sunflowers, and spreadness of coverings.

• Expectation Threshold Theorem: Similarly, the expectation threshold also governs the true threshold within a logarithmic factor:

$$p_c(\mathcal{F}) \le K'\, T_{\rm exp}(\mathcal{F})\, \log \ell$$

(Park et al., 2022).

• Talagrand’s Constant-Factor Conjecture: Talagrand conjectured the existence of a universal $L$ such that

$$T_{\rm frac}(\mathcal{F}) \le L\, T_{\rm exp}(\mathcal{F})$$

for all monotone nontrivial $\mathcal{F}$ (Fischer et al., 2023, Fischer et al., 27 May 2025). The general conjecture is open, but substantial special cases and random support settings are now resolved (Fischer et al., 21 Oct 2025).

3. Key Methods: Fractional Covers, LP Duality, and Selector Processes

The core framework is the use of linear programming to model set-coverings. The fractional expectation threshold emerges as the optimal value of the fractional LP relaxation, which can often be analyzed using LP duality:

• Primal LP (fractional covering): assign weights $q: 2^X \to [0,1]$ so that for every minimal forbidden structure $M$,

$$\sum_{B \subseteq M} q(B) \ge 1, \quad \sum_{B \subseteq X} q(B) p^{|B|} \le 1$$

(Friedgut et al., 2013, Pham, 4 Dec 2024).

• Dual LP: maximizes over certain “packing” variables with constraints reflecting no ground element is overused, facilitating tight probabilistic lower bounds.

Selector process inequalities—probabilistic symmetrizations spanning “witnesses” of fractional covers—have become instrumental in reduction and rounding arguments and were crucial for resolving the Kahn–Kalai and bounded-support cases of Talagrand (Dubroff et al., 1 Dec 2024, Pham, 4 Dec 2024).

4. Special Cases, Applications, and Random Support Regimes

Several important properties admit explicit computation of $T_{\rm frac}$ and comparison to $(p_c, T_{\rm exp})$:

• Perfect Matchings in $r$-uniform Hypergraphs: $T_{\rm frac} \sim n^{-(r-1)}$, and $p_c = O(n^{-(r-1)} \log n)$. The log gap is tight (Frankston et al., 2019).
• Bounded Degree Spanning Trees/Graphs: Fractional spreadness estimates govern $T_{\rm frac}$, leading to sharp or nearly sharp thresholds (Frankston et al., 2019).
• Axial Multi-dimensional Assignments: The spreadness and LP-embedding give $Z = O(n^{-(d-2)})$ for minimum assignment value (Frankston et al., 2019).
• Random Support Model: For a random $k$-uniform hypergraph $H^{(k)}(n,m)$, with $g = (1/r)\mathbf{1}_H$, $w(g,p) = m p^k / r$, and $p = (r/m)^{1/k}$. A constant-factor relation

$T_{\rm frac}(F) \le L\, T_{\rm exp}(F)$

holds a.a.s. for $L = 4e^3$ (Fischer et al., 21 Oct 2025). This average-case result validates Talagrand’s conjecture for random supports.

5. Rounding Fractional Solutions and Bounded-Size Supports

A series of advances resolve Talagrand’s conjecture for fractional covers supported on small sets:

• Sharp Selector Process and Rounding: If a fractional cover $w$ is supported on sets of size $\le t$ and $w(g,p) \le 1/2$, then there exists an integral cover at $q = c p / \log t$ ($c$ absolute), giving $T_{\rm int}(H) = \Theta(T_{\rm frac}(H))$ for bounded $t$ (Pham, 4 Dec 2024, Dubroff et al., 1 Dec 2024).
• This methodology relies on a version of the selector process lemma, allowing passage from fractional to integral coverings via random sampling and probabilistic inequalities.

6. Extensions, Limitations, and Outstanding Open Problems

Recent progress includes:

• Complete resolution for many hypergraph clique and linear support cases, via precise overlap control and randomized selection mechanisms (Fischer et al., 27 May 2025, Fischer et al., 2023).
• Bounded-support: For fractional covers on sets of bounded cardinality, the integral and fractional thresholds are within an absolute constant (Pham, 4 Dec 2024, Dubroff et al., 1 Dec 2024).
• Average-case bounds: Random support (hypergraph) models yield constant-factor relations with high probability (Fischer et al., 21 Oct 2025).

However, the general constant-factor conjecture for arbitrary monotone properties and arbitrary supports remains open. Intermediate complexity cases—e.g., where the underlying support hypergraph is neither sufficiently random, bounded-codegree, nor of very high uniformity—do not yet admit a universal rounding scheme or LP reduction (Fischer et al., 2023).

Open directions include:

• Generalizing the random sampling/overlap control methods to all support hypergraphs.
• Removing logarithmic factors in threshold approximation for broad families.
• Identifying universal combinatorial structures (e.g., containers, sunflowers, Lovász Local Lemma frameworks) that could encompass all hard cases (Fischer et al., 2023, Fischer et al., 27 May 2025).

7. Significance in Threshold Theory and Probabilistic Combinatorics

The fractional expectation threshold provides a unifying framework, subsuming a variety of previously ad hoc arguments for threshold bounds in random graphs, hypergraphs, and combinatorial designs. By characterizing the maximum fractional “budget” needed to cover a monotone property at prescribed cost, it both sharpens and quantitatively explains logarithmic gaps in threshold behavior, simplifies upper bound derivations, and underpins the resolution of longstanding conjectures. Its LP-based nature facilitates connections to algorithmic rounding (randomized and combinatorial), probabilistic embedding results, and the structure of extremal combinatorial families (Frankston et al., 2019, Park et al., 2022, Friedgut et al., 2013, Fischer et al., 2023).

The concept is now essential in combinatorial probability, random discrete structures, and algorithmic extremal combinatorics, with tools from LP-duality, coupling, probabilistic inequalities, and hypergraph container/balanced allocation models converging in their analysis and application.