Let $k>1$ be a positive integer. A set $S{}$ is called [i]good[/i] if there exists a colouring of the positive integers with $k{}$ colours, such that no element from  $S{}$ can be written as the sum of two distinct positive integers having the same colour. Find the greatest positive integer $t{}$ (in terms of $k{}$) for which the set \[S=\{a+1,a+2,\ldots,a+t\}\]is good, for any positive integer $a{}$.
1. **Understanding the Problem:**
   We need to find the greatest positive integer \( t \) such that the set \( S = \{a+1, a+2, \ldots, a+t\} \) is *good* for any positive integer \( a \). A set \( S \) is called *good* if there exists a coloring of the positive integers with \( k \) colors such that no element from \( S \) can be written as the sum of two distinct positive integers having the same color.

2. **Simplifying the Problem:**
   Notice that the problem becomes simpler if we consider the smallest possible \( a \), which is \( a = 1 \). If we can find \( t \) for \( a = 1 \), it will work for any larger \( a \) because increasing \( a \) makes it less likely for sums to fall within the set \( S \).

3. **Setting \( a = 1 \):**
   For \( a = 1 \), the set \( S \) becomes \( \{2, 3, 4, \ldots, t+1\} \). We need to ensure that no element in \( S \) can be written as the sum of two distinct elements of the same color.

4. **Analyzing the Pairs:**
   Consider the pairs \((x, y)\) such that \( x + y = z \in S \). For \( x = 2 \), the possible pairs are \((2, y)\) where \( y \in \{3, 4, \ldots, t-1\} \). This gives us \( t-2 \) pairs. For \( x = 3 \), the possible pairs are \((3, y)\) where \( y \in \{4, 5, \ldots, t-2\} \), giving us \( t-4 \) pairs, and so on.

5. **Counting the Pairs:**
   For each \( x = 2 + i \), the number of pairs is \( t - 2i - 1 \). We need to ensure that each pair \((x, y)\) has different colors. This means we need at least \( t - 2i \) colors for each \( i \).

6. **Finding the Maximum \( t \):**
   We need to find the maximum \( t \) such that we can color the set \( S \) with \( k \) colors. The process continues until \( i = j \) where \( j \leq k \) and \( t - 2j - 1 > 0 \). This gives us the inequality:
   \[
   t \leq 2k - 2
   \]
   If \( t > 2k - 2 \), we would need more than \( k \) colors, which is not allowed.

7. **Conclusion:**
   Therefore, the greatest positive integer \( t \) for which the set \( S = \{a+1, a+2, \ldots, a+t\} \) is good for any positive integer \( a \) is \( t = 2k - 2 \).

The final answer is \( \boxed{2k - 2} \)