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- W3043928474 abstract "A set $A$ of natural numbers possesses property $mathcal{P}_h$, if there are no distinct elements $a_0,a_1,dots ,a_hin A$ with $a_0$ dividing the product $a_1a_2dots a_h$. ErdH{o}s determined the maximum size of a subset of ${1,ldots, n}$ possessing property $mathcal{P}_2$. More recently, Chan, GyH{o}ri and S'arkozy solved the case $h=3$, finally the general case also got resolved by Chan, the maximum size is $pi(n)+Theta_h(frac{n^{2/(h+1)}}{(log n)^{2}})$. In this note we consider the counting version of this problem and show that the number of subsets of ${1,ldots, n}$ possessing property $mathcal{P}_h$ is $T(n)cdot e^{Theta(n^{2/3}/log n)}$ for a certain function $T(n)approx (3.517dots)^{pi(n)}$. For $h>2$ we prove that the number of subsets possessing property $mathcal{P}_h$ is $T(n)cdot e^{sqrt{n}(1+o(1))}$. This is a rare example in which the order of magnitude of the lower order term in the exponent is also determined." @default.
- W3043928474 created "2020-07-29" @default.
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- W3043928474 date "2020-12-01" @default.
- W3043928474 modified "2023-09-27" @default.
- W3043928474 title "The counting version of a problem of Erdős" @default.
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- W3043928474 doi "https://doi.org/10.1016/j.ejc.2020.103187" @default.
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