Let an integer $ n > 3$. Denote the set $ T\equal{}\{1,2, \ldots,n\}.$ A subset S of T is called [i]wanting set[/i] if S has the property: There exists a positive integer $ c$ which is not greater than $ \frac {n}{2}$ such that $ |s_1 \minus{} s_2|\ne c$ for every pairs of arbitrary elements $ s_1,s_2\in S$. How many does a [i]wanting set[/i] have at most are there ?
1. **Define the problem and the set \( T \):**
   Let \( n \) be an integer greater than 3. Define the set \( T = \{1, 2, \ldots, n\} \). A subset \( S \) of \( T \) is called a *wanting set* if there exists a positive integer \( c \) such that \( c \leq \frac{n}{2} \) and for every pair of elements \( s_1, s_2 \in S \), \( |s_1 - s_2| \neq c \).

2. **Determine the maximum size of a wanting set:**
   We need to find the maximum size of such a subset \( S \). Assume \( S \) is a wanting set with the maximum number of elements. We can assume that at least half of the elements of \( S \) are less than or equal to \( \frac{n}{2} \). Let this subset be \( T \).

3. **Analyze the elements in \( T \):**
   For every \( t \leq \frac{n}{2} \), the sets \( S \) and \( T + t \) (where \( T + t = \{x + t \mid x \in T\} \)) must be disjoint. This is because if \( S \) and \( T + t \) were not disjoint, there would exist elements \( s_1, s_2 \in S \) such that \( |s_1 - s_2| = t \), which contradicts the definition of a wanting set.

4. **Calculate the upper bound for the size of \( S \):**
   Since \( S \) and \( T + t \) are disjoint for every \( t \leq \frac{n}{2} \), we have:
   \[
   \frac{3|S|}{2} \leq n
   \]
   This inequality arises because \( S \) can be divided into two parts, each containing at most \( \frac{|S|}{2} \) elements, and the union of these parts and their shifted versions must fit within the set \( T \).

5. **Solve the inequality:**
   \[
   \frac{3|S|}{2} \leq n \implies |S| \leq \frac{2n}{3}
   \]

6. **Verify the result with an example:**
   Consider the set \( S = \{1, 2, \ldots, \left\lfloor \frac{n}{3} \right\rfloor, \left\lfloor \frac{2n}{3} \right\rfloor + 1, \ldots, n\} \). This set satisfies the condition that for every pair of elements \( s_1, s_2 \in S \), \( |s_1 - s_2| \neq c \) for any \( c \leq \frac{n}{2} \).

Thus, the maximum size of a wanting set is \( \left\lfloor \frac{2n}{3} \right\rfloor \).

The final answer is \( \boxed{ \left\lfloor \frac{2n}{3} \right\rfloor } \)