Suppose $X \subseteq \lbrace 0 , 1 \rbrace ^{m}$ such that $|X| \geq 2^{0.8m}$, and $m \geq 2$, then prove that there exists $x,y \in X$ with $||x - y||_{1} \geq m/2$.

My approach to prove this was if there is no such $x,y$, then $X$ is inside a hamming ball of radius $m/2$ . But this does not give me a tight enough inequality on the size of $X$.

Suppose $X \subseteq \lbrace 0 , 1 \rbrace ^{m}$ such that $|X| \geq 2^{0.8m}$, and $m \geq 2$, then prove that there exists $x,y \in X$ with $||x - y||_{1} \geq m/2$.

My approach to prove this was if there is no such $x,y$, then $X$ is inside a Hamming ball of radius $m/2$ . But this does not give me a tight enough inequality on the size of $X$.