Covering systems with the sum of the reciprocals of the moduli close to 1

In 1952, H. Davenport posed the problem of determining a condition on the minimum modulus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m 0"> <mml:semantics> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">m_{0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a finite distinct covering system that would imply that the sum of the reciprocals of the moduli in the covering system is bounded away from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In 1973, P. Erdős and J. Selfridge indicated that they believed that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m 0 greater-than 4"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m_{0} &gt; 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> would suffice. We provide a proof that this is the case.