On a conjecture of Erd\H{o}s

Let $\mathcal{P}$ denote the set of all primes. In 1950, P. Erd\H{o}s conjectured that if $c$ is an arbitrarily given constant, $x$ is sufficiently large and $a_1,\dots ,a_t$ are positive integers with $a_1<a_2<\cdot \cdot \cdot <a_t⩽x$ and $t>\log x$, then there exists an integer $n$ so that the number of solutions of $n=p+a_i$ $(p\in $\mathcal{P}$, 1\le i\le t)$isgreaterthan$c$. In this note, we confirm this old conjecture of Erd\H{o}s.