Determine the largest integer $n\geq 3$ for which the edges of the complete graph on $n$ vertices
can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.
1. **Define the problem and initial assumptions:**
   We need to determine the largest integer \( n \geq 3 \) for which the edges of the complete graph \( K_n \) can be assigned pairwise distinct non-negative integers such that the edges of every triangle form an arithmetic progression. 

2. **Translate and scale the assignments:**
   We can assume that the largest number assigned to an edge is 1 and the smallest is 0. This is possible by translating and scaling the assignments.

3. **Case 1: Non-incident edges:**
   Suppose the edges with numbers 0 and 1 are not incident. Consider a tetrahedron (complete graph on 4 vertices). We need to check if the edges can form arithmetic progressions. 
   
   - Assign numbers to the edges such that they form arithmetic progressions. 
   - If the edges are \(0, 1, \frac{1}{3}, \frac{2}{3}\), some of the numbers must be equal, which contradicts the requirement for distinct numbers.

4. **Case 2: Incident edges:**
   Suppose the edges with numbers 0 and 1 are incident, say \(0 \to AB\) and \(1 \to AC\). Then, the edge \(BC\) must be assigned \(\frac{1}{2}\) to form an arithmetic progression with \(AB\) and \(AC\).

5. **Consider another vertex \(X\):**
   - Assign numbers to the edges \(XA, XB, XC\) such that they form arithmetic progressions with the existing edges.
   - Two possible assignments are:
     1. \(XA = \frac{1}{5}, XB = \frac{2}{5}, XC = \frac{3}{5}\)
     2. \(XA = \frac{5}{7}, XB = \frac{5}{14}, XC = \frac{3}{7}\)

6. **Implication for \(n \leq 5\):**
   - If there are three vertices \(X, Y, Z\) apart from \(A, B, C\), some of the assignments must be the same, which contradicts the requirement for distinct numbers.
   - Assume \(n = 5\) with vertices \(A, B, C, X, Y\). Assign numbers to the edges involving \(X\) and \(Y\):
     - \(XA = \frac{1}{5}, XB = \frac{2}{5}, XC = \frac{3}{5}\)
     - \(YA = \frac{5}{7}, YB = \frac{5}{14}, YC = \frac{3}{7}\)
   - Check if there is a number to be assigned to \(XY\) to satisfy the requirements. An easy calculation shows that no such number exists.

7. **Conclusion:**
   - The largest \(n\) for which the edges of the complete graph \(K_n\) can be assigned pairwise distinct non-negative integers such that the edges of every triangle form an arithmetic progression is \(n = 4\).

The final answer is \( \boxed{ n = 4 } \)