A subset $S$ of the natural numbers is called dense for every $7$ consecutive natural numbers, at least $5$ of them are in $S$. Show that there exists a dense subset for which the equation $a^2+b^2=c^2$ has no solution for $a,b,c \in S$.
Problem
Source: Kosovo Math Olympiad 2025, Grade 10, Problem 3
Tags: Kosovo, national olympiad, number theory, Sets