Given an integer $n\geq 3$. For each $3\times3$ squares on the grid, call this $3\times3$ square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white. 

Now suppose one can paint an infinite grid by white or black, so that one can select an $a\times b$ rectangle which contains at least $n^2-n$ isolated $3\times 3$ square. Find the minimum of $a+b$ that such thing can happen.

(Note that $a,b$ are positive reals, and selected $a\times b$ rectangle may have sides not parallel to grid line of the infinite grid.)
1. **Define Isolated Centers and Set \( S \)**:
   - Call the center square of an isolated \(3 \times 3\) square an isolated center.
   - Consider the set \( S \) of isolated centers in the \(a \times b\) rectangle.
   - Suppose a \(a \times b\) rectangle contains \(n^2 - n\) isolated squares, and take one with minimum \(a \times b\).

2. **Partition the Plane**:
   - For a grid square \(X\) located at \((x_0, y_0)\), partition the plane into five parts: \(\{X\}\) and \(A^+(X), B^+(X), A^-(X), B^-(X)\).
   - Define:
     \[
     A^+(X) = \{(x, y) \mid x + y > x_0 + y_0, x - y \geq x_0 - y_0\}
     \]
     \[
     B^+(X) = \{(x, y) \mid x + y \geq x_0 + y_0, x - y < x_0 - y_0\}
     \]
     \[
     A^-(X) = \{(x, y) \mid x + y < x_0 + y_0, x - y \leq x_0 - y_0\}
     \]
     \[
     B^-(X) = \{(x, y) \mid x + y \leq x_0 + y_0, x - y > x_0 - y_0\}
     \]
   - Note that \(X \in A^+(Y) \Leftrightarrow Y \in A^-(X)\) and similarly for \(B\).

3. **Define Order and Chains**:
   - Define an order \((S, \prec)\) such that \(X \prec Y \Leftrightarrow Y \in A^+(X)\).
   - Consider the maximum length chain \(U\) and maximum length antichain \(V\). Let their lengths be \(p\) and \(q\) respectively.
   - Note that \(|U \cap V|\) is at most 1 since \(V\) doesn't have any two comparable elements.

4. **Define Sets \(Y^+, Y^-, X^+, X^-\)**:
   - Define:
     \[
     Y^+ = \bigcup_{i=0}^{p} A^+(U_i) \cap B^+(U_{i+1})
     \]
     \[
     Y^- = \bigcup_{i=0}^{p} A^+(U_i) \cap B^-(U_{i+1})
     \]
     \[
     X^+ = \bigcup_{i=0}^{q} B^+(V_i) \cap A^+(V_{i+1})
     \]
     \[
     X^- = \bigcup_{i=0}^{q} B^+(V_i) \cap A^-(V_{i+1})
     \]
   - These sets partition \(S \setminus U\) and \(S \setminus V\).

5. **Step 1: Area Calculation**:
   - Such \(a \times b\) rectangle contains at least \(4n^2 - 2\) unit grid squares.
   - Place disjoint lattice polygons inside the \(a \times b\) rectangle so that the sum of their areas is \(4n^2 - 2\) or more.
   - Place tiles as described in the problem statement.

6. **Step 2: \(a + b \geq 4n\) for Rectangles Parallel to Grid**:
   - Assume the rectangle has lattice points on each side by minimality.
   - \(a\) and \(b\) are integers and \(ab \geq 4n^2 - 2\).
   - Suppose \(a + b \leq 4n - 1\). Then \(ab \leq 2n(2n - 1) < 4n^2 - 2\), which is a contradiction.
   - Therefore, \(a + b \geq 4n\) in this case.

7. **Step 3: \(ab \geq 4n^2\) for Rectangles Not Parallel to Grid**:
   - Assume the rectangle touches the polygon in Step 1 on each side.
   - Use trigonometry to show that the minimum occurs on the boundary.
   - Consider the cases where the slope is \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}\) and show that the additional area is at least 2 in all cases.
   - Therefore, \(ab \geq 4n^2\) and by AM-GM, \(a + b \geq 2\sqrt{ab} = 4n\).

8. **Conclusion**:
   - Finally, we get \(a + b \geq 4n\) in all cases.
   - As an example, color \((x, y)\) black iff \(2 \mid x\) and \(2 \mid y\), then we can select a \((2n - 1) \times (2n + 1)\) rectangle containing \(n^2 - n\) isolated squares.

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