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Source. Conjecture 2 and its remark, printed p. 390 (PDF p. 10).

Conjectural input. There is one sequence of positive constants $t(1),t(2),\ldots$ with

$$\log t(j)=o(j\log j)\qquad(j\to\infty)$$

such that every nonempty finite intersecting family $\mathcal A$ of distinct finite nonempty sets has a nonempty set $C$ satisfying

$$\#\{S\in\mathcal A:C\subseteq S\} \ge\frac{|\mathcal A|}{t(|C|)}. \tag{1}$$

The sequence is universal: it does not depend on the family, its ground set or its maximum member size. Intersecting means that every two members meet. The nonempty family convention excludes a vacuous case and is the one used in the progression argument. Enlarging $t(j)$ to $\max\{1,t(j)\}$ preserves the assertion and its growth condition, so the proofs may assume $t(j)\ge1$.

The complete conditional Theorem 2 proof uses exactly (1). Distinct residual prime supports are essential: the hypothesis concerns a family of sets, not a multiset with arbitrary weights. The pruning and full-block invariants supply that distinctness.

The source remarks that a weaker sequence with $t(j)\ll j^{j+2}$ can be proved, without supplying that proof. This remark is retained only as a source statement and is not used here. The near-sharpness example for Lemma 3.4 cited from Erdős–Lovász is likewise external historical context. No current theorem establishing (1), and no equivalence with a later sunflower formulation, is asserted by this page.