OFM 2021 J3

Every point in the plane was colored in red or blue. Prove that one the two following statements is true: $\bullet$ there exist two red points at distance $1$ from each other; $\bullet$ there exist four blue points $B_1$ , $B_2$ , $B_3$ , $B_4$ such that the points $B_i$ and $B_j$ are at distance $|i - j|$ from each other, for all integers $i$ and $j$ such as $1 \le i \le 4$ and $1 \le j \le 4$ .