Erdős Problem #1188

Erdős Problem 1188 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

A distinct covering system consists of residue classes $a_i\pmod{n_i}$ with distinct integers $1<n_1<\cdots<n_k$, whose union covers $\mathbb Z$. It is minimal when deleting any nonempty collection of its classes destroys that covering property. Let $F(x)$ count these minimal systems whose moduli are at most $x$. Determine the growth of $F(x)$.

CONTEXT AND SCOPE

This replaces the earlier statement-only record with an elementary mathematical attempt. The arguments below establish only their stated partial results; no novelty or full solution is claimed.

Systems include the chosen residue classes; this differs from counting only sets of moduli. Residues are understood modulo their moduli.

ATTACK PLAN AND WORK

For integral $X\ge2$, each modulus $m\in[2,X]$ is absent or has one of its $m$ residues selected. Ignoring coverage and minimality gives the explicit upper bound \[F(X)\le\prod_{m=2}^X(m+1)=\frac{(X+1)!}{2}.\tag{1}\] There is a finite exact minimality certificate. Work modulo the least common multiple $Q$ of the chosen moduli. A covering is minimal if and only if every class has a residue of $\mathbb Z/Q\mathbb Z$ covered by that class alone. Necessity follows by deleting that class if it has no such witness; sufficiency follows because deleting any class loses its witness.

For example the five classes \[0\pmod2,\quad0\pmod3,\quad1\pmod4,\quad5\pmod6,\quad7\pmod{12}\] cover all residues modulo 12. Private witnesses, in the listed order, are $2,3,1,11,7$. This proves minimality directly. Translating all classes by each of the twelve residues gives twelve different minimal systems, since the modulus-12 residue distinguishes them. Thus $F(12)\ge12$ under the source's convention of counting residue choices, without identifying translations.

This certificate tests minimality of a particular assignment. It must not be confused with irreducibility of its moduli, which also rules out different assignments to proper subsets and is addressed separately in problem 1189.

FINITE COMPUTATIONAL CHECK

Direct enumeration of the twelve residue positions verifies coverage, each of the five private witnesses, and distinctness of the twelve translated systems.

VERIFICATION AND REMAINING GAP

Estimate the total number on a sharp scale as $X$ grows. The factorial upper bound and the explicit twelve examples are elementary benchmarks and do not recover the stronger growth bounds cited on the official page.

SOURCES

https://www.erdosproblems.com/1188

FINAL

UNRESOLVED. The full source problem remains outside the scope of the proved partial results.

COMPLETION ESTIMATE

$10\%$. This is a conservative subjective estimate toward the whole problem, not a probability that the argument is correct or a claim that a fixed fraction of the problem has been solved.