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Source. Sections 1–4 of de la Bretèche–Ford–Vandehey (2013), printed pp. 381–389 (PDF pp. 1–9). This page records notation and external inputs, not proofs of those inputs.

For real $x$ tending to infinity, put

$$X=\log x,\qquad \ell=\log X,\qquad B=\sqrt{X/\ell},\qquad T=\sqrt{X\ell}=B\ell,\qquad L(\alpha,x)=e^{\alpha T}.$$

All logarithms are natural. Define $f(x)$ to be the maximum cardinality of a set $\mathcal Q\subseteq\{1,\ldots,\lfloor x\rfloor\}$ for which one can choose residues $a_q$ so that the classes $a_q\pmod q$ are pairwise disjoint. The maximum exists: there are finitely many possible sets and, for each set, finitely many choices of residues modulo its members. This agrees with the source's supremum over admissible sets of positive integers, after restriction to $[1,x]$.

Distinct moduli are part of this definition. The source allows modulus one; it can occur only in a singleton family. The asymptotic lower construction has size tending to infinity, so an extremal family for large $x$ contains no modulus one.

For a positive integer $n$, write

$$\omega(n)=\#\{p:p\mid n\},\qquad \Omega(n)=\sum_{p^\nu\parallel n}\nu,\qquad \operatorname{ker}(n)=\prod_{p\mid n}p,\qquad h(n)=\prod_{p^\nu\parallel n}\nu.$$

The empty products at $n=1$ are one; $\omega(1)=\Omega(1)=0$.

Prime estimates

The analytic external input is the classical prime number theorem

$$\pi(u)=(1+o(1))\frac{u}{\log u}\qquad(u\to\infty).$$

In particular, $\pi(2u)-\pi(u)=(1+o(1))u/\log u$. These estimates hold uniformly for all real $u\ge u_0(x)$ whenever $u_0(x)\to\infty$: this is the usual eventual meaning of the one-variable limit. The lower construction uses them to count primes in its intervals and to find a common prime in $[y_0,2y_0]$. Lemma 3.2 uses the same theorem to estimate $\pi(\sqrt{X}/5)$. No short-interval theorem or effective error term is assumed. The analytic proof of the prime number theorem is external.

Congruences

We use unique prime factorization and the finite Chinese remainder theorem. In particular, prescribed residues modulo pairwise coprime integers specify one residue modulo their product. The generalized two-modulus form says that

$$a\pmod m\quad\hbox{and}\quad b\pmod n \quad\hbox{intersect if and only if}\quad a\equiv b\pmod{\gcd(m,n)}.$$

Thus agreement modulo all already selected full prime-power blocks forces a disjoint pair to share a prime outside those blocks. Agreement only modulo the product of the underlying primes would not suffice.

The Euler-product and Rankin estimates needed below are proved in the individual lemma pages. They use only convergent nonnegative series, finite counting and elementary integral bounds. The source's reference to Tenenbaum for Rankin's method is not an omitted same-paper argument. The Erdős–Lovász-style set bound used here is likewise fully proved in Lemma 3.4.

The sequence assumption in Conjecture 2 is an additional conditional input for Theorem 2 only. It is not used for the unconditional Theorem 1, and no later sunflower theorem is substituted into the original proof.