ILL 1986 Problem 15

Let $\mathbb N = B_1\cup\cdots \cup B_q$ be a partition of the set $\mathbb N$ of all positive integers and let an integer $l \in \mathbb N$ be given. Prove that there exist a set $X \subset \mathbb N$ of cardinality $l$ , an infinite set $T \subset \mathbb N$ , and an integer $k$ with $1 \leq k \leq q$ such that for any $t \in T$ and any finite set $Y \subset X$ , the sum $t+ \sum_{y \in Y} y$ belongs to $B_k.$