RU-SPB 2010 Problem 117

$600$ integer numbers from $[1,1000]$ colored in red. Natural segment $[n,k]$ is called yummy if for every natural $t$ from $[1,k-n]$ there are two red numbers $a,b$ from $[n,k]$ and $b-a=t$ . Prove that there is yummy segment with $[a,b]$ with $b-a \geq 199$