Talagrand Conjecture and Thresholds

• Talagrand Conjecture is defined as the claim that expectation and fractional expectation thresholds are universally within a constant factor for increasing families in probability.
• It employs combinatorial decompositions, LP formulations, and selector process theorems to round fractional covers to integral covers in discrete settings.
• While concrete bounds are established for special cases, extending these results to general unbounded support remains an open challenge.

The Talagrand Conjecture spans several domains in probability, combinatorics, analysis, and topology. Most prominently, in modern probabilistic combinatorics, it refers to the claim that expectation and fractional expectation thresholds are universally within a constant factor—transforming the landscape of random discrete structures, threshold phenomena, and integer/fractional covering duality.

1. Integral and Fractional Expectation Thresholds

Let $X$ be a finite set and $\mathcal F \subseteq 2^X$ an increasing family (i.e., $A \in \mathcal F$, $A \subseteq B$ implies $B \in \mathcal F$). Consider the random $p$-subset $X_p$ (each $x$ included independently with probability $p$), and define:

• Threshold $p_c(\mathcal F)$: $$p_c(\mathcal F) = \inf \{ p : \Pr[X_p \in \mathcal F] \ge 1/2 \}$$.
• Expectation threshold $q(\mathcal F)$: The largest $p$ such that there is a family $G \subseteq 2^X$ with $\mathcal F \subseteq \langle G \rangle$ and $\sum_{S \in G} p^{|S|} \le 1/2$, where $\langle G \rangle$ is the up-closure of $G$.
• Fractional expectation threshold $q_f(\mathcal F)$: The largest $p$ such that there exists $g : 2^X \to [0,1]$ with $\mathcal F \subseteq \langle g \rangle$ and $\sum_{S \subseteq X} g(S) p^{|S|} \le 1/2$.

By definition, $$q(\mathcal F) \leq q_f(\mathcal F) \leq p_c(\mathcal F)$$. The threshold $q_f(\mathcal F)$ arises from the fractional relaxation of the associated integer linear program (LP), while $q(\mathcal F)$ corresponds to the integer program (Pham, 4 Dec 2024).

2. Formal Statements of the Talagrand Conjecture(s)

Global Conjecture (Frankston et al., 2021, Frankston et al., 2019, Pham, 4 Dec 2024, Fischer et al., 27 May 2025):

There is a universal constant $L < \infty$ such that, for every finite $X$ and every increasing family $\mathcal F \subseteq 2^X$, $q_f(\mathcal F) \leq L \, q(\mathcal F)$ or equivalently, every small fractional cover can be rounded to an integral cover at a loss of at most a constant factor in $p$.

Logarithmic Variant (Frankston et al., 2019):

There is an absolute constant $K$ such that, for all $X, \mathcal F$, $$p_c(\mathcal F) \leq K \, q_f(\mathcal F) \log \ell(\mathcal F)$$ with $\ell(\mathcal F)$ the size of the largest minimal member of $\mathcal F$.

Special Cases and Bounded-Support Version (Frankston et al., 2021, Dubroff et al., 1 Dec 2024, Pham, 4 Dec 2024):

• For fractional solutions supported only on sets of size at most $t$, the integral threshold increases by at most a $O(\log t)$ factor:

$$p_E(\mathcal H) \leq C\, p_f(\mathcal H) \leq C' p_E(\mathcal H) \log t$$

3. Proof Strategies, Resolved Cases, and Technical Ingredients

Top-level strategies invoke combinatorial decompositions, probabilistic covering arguments, and the powerful "selector process" paradigm (Park et al., 2022, Pham, 4 Dec 2024, Dubroff et al., 1 Dec 2024):

• Duality and LP Formulation: The problem is cast in terms of covering integer/fractional LPs associated to the system $$\forall H \in \mathcal H, \sum_{W \subset H} w_W \geq 1$$ (Pham, 4 Dec 2024, Fischer et al., 27 May 2025).
• Selector Process Theorem: A sharp concentration result for selector processes shows that, if a family fails to have a small $p$-cover, then a random thinning selects a constant fraction of mass (in the sense of linear functionals of the cover weights), yielding explicit rounding schemes from fractional to integral covers (Pham, 4 Dec 2024, Dubroff et al., 1 Dec 2024, Park et al., 2022).
• Special Cases: The conjecture is resolved for:
  • Singleton-support: Trivial case; Talagrand's original covering-by-prefixes proof (Frankston et al., 2021).
  • Pair-support: All $\lambda$ supported on 2-element sets (graph-theoretic setting), via weighted/unweighted combinatorial covering and star-forest decompositions (Frankston et al., 2021).
  • Bounded-size support: For fractional covers supported on subsets of size at most $t$, the $O(\log t)$ bound is tight and explicit (Pham, 4 Dec 2024, Dubroff et al., 1 Dec 2024).
  • Random and pseudorandom support: For fractional weights assigned to random uniform $k$-uniform hypergraphs, a.a.s. an $L$-factor rounding suffices (Fischer et al., 21 Oct 2025, Fischer et al., 27 May 2025).
  • Clique hypergraph cases: For uniform cliques, the DeMarco–Kahn random-union method generalizes the reduction (Fischer et al., 27 May 2025).

Table: Key Resolved Cases

Support of Fractional Solution	Constant Factor Holds?	Reference
Singletons	Yes, trivial	(Frankston et al., 2021)
Pairs (edges)	Yes, explicit $L$	(Frankston et al., 2021)
Sets of size $\leq t$	$O(\log t)$	(Pham, 4 Dec 2024)
Random/uniform hypergraphs	Yes, a.a.s.	(Fischer et al., 21 Oct 2025)
General/unbounded	Open	—

For bounded-support, the argument leverages a sharp selector-process theorem (Pham) and entropy-compression via "towers of fragments" to perform effective rounding while controlling the blowup in $p$ (Pham, 4 Dec 2024).

4. Selector Process Conjecture and Its Resolution

Central to several advances is the Talagrand Selector Process Conjecture, formulated analogously to the majorizing measure theory for Gaussian suprema (Park et al., 2022):

• Bernoulli-$p$ Selector Process Theorem: For any family $\mathcal A$ of weight vectors on $X$, the set $$\{ S \subset X : \sup_{a \in \mathcal A} \sum_{i \in S} a_i > L \mathbb E[\sup_a \sum_{i \in X_p} a_i] \}$$ can be covered by up-sets whose total $p$-measure is at most $1/2$, with $L$ universal (Park et al., 2022).
• Implications: This result directly yields the bounded-support rounding theorems as "fractional $p$-smallness for sets of size $\leq t$ implies integral $(p/\log t)$-smallness" (Dubroff et al., 1 Dec 2024, Pham, 4 Dec 2024).

5. Connections to Isoperimetric Inequalities and Correlation Bounds

The conjecture interfaces with numerous threads in Boolean analysis:

• Isoperimetry: Talagrand’s isoperimetric inequalities on the hypercube leverage moments of sensitivity; the progression from the Margulis and KKL inequalities to tight, $\ell_2$-sensitive boundary bounds involved similar multi-scale and variance/influence-based arguments (Eldan et al., 2022).
• Correlation inequalities: Talagrand's quantitative Harris–Kleitman program lower-bounds covariance $\operatorname{Cov}(f, g)$ in terms of coordinate influences; its exact sharp form is established in submodular/supermodular regimes (Chang et al., 25 Oct 2025).

6. Unresolved Directions and Open Problems

• General Unbounded-Support Case: The full conjecture for arbitrary fractional covers remains open, with the best known guarantee being a $\log |X|$-factor (Pham, 4 Dec 2024).
• Sharp Constants, Pseudorandomness: The optimal value of $L$, and the combinatorial or spectral conditions under which the constant-factor rounding works for general supports, are not determined (Fischer et al., 21 Oct 2025).
• Extension of Selector Bounds: Further strengthening or variant selector process concentration inequalities, or entropy compression arguments in the spirit of "fragments" and "towers," may bridge the gap for the general case (Dubroff et al., 1 Dec 2024, Pham, 4 Dec 2024).
• Applications: The conjecture is intimately tied to the tightness of random thresholds for matchings, random assignment problems, and hypergraph property thresholds (Frankston et al., 2019).

7. Broader Context and Related Conjectures

The Talagrand Conjecture sits at the nexus of threshold phenomena, extremal combinatorics, probabilistic covering duality, LP rounding in discrete geometry, and even theoretical computer science (notably, algorithmic derandomization and CSP sharp thresholds). Analogous paradigms appear in:

• Gaussian process theory: Majorizing measure theorem and chaining techniques (Park et al., 2022).
• Empirical process theory: Empirical suprema, deviation inequalities, half-space coverings, and chaining (Park et al., 2022).
• Guerra–Talagrand bounds: In high-dimensional spin glass models, similar duality and interpolation principles underpin the Parisi formula and its ultrametricity/chaos consequences (Chen, 2015).

A plausible implication is that further integration of probabilistic chaining, pseudorandomness, and selector-process tools will be decisive for resolving the general conjecture.

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References:

• (Frankston et al., 2021) "On a problem of M. Talagrand"
• (Frankston et al., 2019) "Thresholds versus fractional expectation-thresholds"
• (Pham, 4 Dec 2024) "A sharp version of Talagrand's selector process conjecture and an application to rounding fractional covers"
• (Dubroff et al., 1 Dec 2024) "Note on a conjecture of Talagrand: expectation thresholds vs. fractional expectation thresholds"
• (Fischer et al., 27 May 2025) "Further remarks on fractional vs. expectation thresholds"
• (Fischer et al., 21 Oct 2025) "Fractional Vs. Expectation Thresholds: Random Support Case"
• (Eldan et al., 2022) "Isoperimetric Inequalities Made Simpler"
• (Chang et al., 25 Oct 2025) "Talagrand-Type Correlation Inequalities for Supermodular and Submodular Functions on the Hypercube"
• (Park et al., 2022) "On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes"
• (Chen, 2015) "Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound"