2019 IMO Shortlist N6

Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geq C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil)

设 $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$，并设 $n$ 为正整数。证明存在常数$C$，使得如果$A \subset\{1,2, \ldots, n\}$ 满足$|A| \geq C \sqrt{n}$，则 A$中存在$a, b \in H$使得$a-b \in H$。 (巴西)