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Source: published version, pp. 236--237, equations (4)--(6), Theorem 1.2, and Conjecture 1.3.

Fractional parameter

An increasing family $\mathcal F\subseteq2^X$ is weakly $p$-small if there is a function $\lambda:2^X\to[0,\infty)$ such that

$$\sum_{S\subseteq F}\lambda_S\ge1 \quad(F\in\mathcal F), \qquad \sum_{S\subseteq X}\lambda_Sp^{|S|}\le\frac12.$$

The fractional expectation threshold is

$$q_f(\mathcal F)=\max\{p:\mathcal F\text{ is weakly }p\text{-small}\}.$$

An ordinary cover supplies a $\{0,1\}$-valued feasible $\lambda$, so $q(\mathcal F)\le q_f(\mathcal F)$. Conversely, for a feasible $\lambda$,

$$\mu_p(\mathcal F) \le \mathbb E\left[\sum_{S\subseteq X_p}\lambda_S\right] =\sum_{S\subseteq X}\lambda_Sp^{|S|} \le\frac12,$$

which gives $q_f(\mathcal F)\le p_c(\mathcal F)$.

Both the published version and arXiv v2 print the domain of $\lambda$ as $2^V$, although no $V$ is defined. The surrounding fixed ground set is $X$; the definition above makes that evident notation correction.

Exact external result and historical conjecture

Theorem 1.2, due to Frankston, Kahn, Narayanan, and Park, states that a universal constant $K$ satisfies

$$p_c(\mathcal F)<Kq_f(\mathcal F)\log\ell(\mathcal F)$$

for every finite $X$ and nontrivial increasing $\mathcal F\subseteq2^X$. This page records that theorem as an exact external result for comparison; its proof is not reconstructed in this source unit.

The paper also records as Conjecture 1.3 Talagrand's proposed universal comparison

$$q(\mathcal F)\ge \frac{q_f(\mathcal F)}{L}.$$

This is historical context as stated in the 2024 source, rather than a claim about the conjecture's later status. Park and Pham prove Theorem 1.1 directly; neither Theorem 1.2 nor Conjecture 1.3 is used in their proof.