THE NUMBER OF MINIMAL COVERING SYSTEMS WITH DISTINCT MODULI

Let $F(x)$ be the number of minimal covering systems of the integers whose moduli are distinct and lie in $[1,x]$. Estimating $F(x)$ is a problem of Erd\H{o}s, known as Erd\H{o}s Problem~\#1188. We determine the order of magnitude of $\log F(x)$:\[ (c_1+o(1))\,x\log x\;\le\;\log F(x)\;\le\;(1-\log 2)\,x\log x+1.5\,x,\]where $c_1=0.0672$. Thus $F(x)=x^{\Theta(x)}$. The upper bound holds for all $x\ge2$. It rests on a local lemma: in a minimal covering system, for every prime $p$, the classes whose moduli are divisible by $p$ project onto all residues modulo the largest power of $p$ dividing a modulus. Hence every modulus of a minimal covering system with distinct moduli at most $x$ is $\sqrt{x}$-smooth. The lower bound comes from a \emph{split construction}. The integers are organised by the least prime power that does not divide them, and many of the resulting classes are split further, along \emph{splitting trees}, according to their residues modulo auxiliary primes. The construction shows that a minimal distinct covering system with moduli at most $x$ can use a positive proportion of all integers up to $x$ as moduli: the largest possible number $k(x)$ of classes satisfies $0.2128\,x\le k(x)\le(1-\log2+o(1))\,x$ for large $x$. In fact, for every $0<h<\frac12$ the construction uses almost all integers $n\le x$ with exactly two prime factors in $(x^{h},\sqrt x]$ and no larger prime factor, and the constants are explicit double integrals of Dickman's function. Splitting only once, by primes from $(x^{1/5},x^{1/4}]$, gives the closed forms $(1-\lambda)\lambda(\lambda-\frac{3}{20})$ and $(1-\lambda)\lambda^{2}$, with $\lambda=\log\frac54$, in place of $c_1$ and $0.2128$, and the systems without splitting give the explicit bound $\log F(x)\ge x/2-3x/\log x$ for all $x\ge40$. The exact values of the constants remain open. Call a modulus \emph{balanced} if none of its prime-power components exceeds its square root. At most $(1+o(1))\,x/\log x$ classes of such a system are unbalanced. An \emph{anchor lemma}, which strengthens the local lemma, shows that the systems with at most $b\ge1$ balanced classes number $\exp\bigl(x/2+O(x\log\log x/\log x+b\log x)\bigr)$. Consequently almost all minimal distinct covering systems with moduli at most $x$ have at least $c\,x$ classes, for any fixed $c<\frac1{45}$. We also compute $F(x)$ exactly for $x\le 24$ and the largest possible number of classes for several $x\le48$.