Question 78
Let $k \gt 1$ be an integer. A set of natural numbers $S$ is called good if all positive integers can be painted in $k$ colors such that no element of $S$ is a sum of two distinct numbers having one and the same color. Find the largest positive integer $t$ for which the set
$$
S = \{a+1, a+2, a+3, \dots, a+t\}
$$
is good for all positive integers $a$.
$$
S = \{a+1, a+2, a+3, \dots, a+t\}
$$
is good for all positive integers $a$.