Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and
$$(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.$$Determine the largest possible number of elements in $A$.
1. **Interpret the Set \( S \)**:
   - The set \( S \) can be visualized as an \( m \times n \) grid where each cell \((a, b)\) corresponds to a point in the grid with \( a \in \{1, 2, \ldots, m\} \) and \( b \in \{1, 2, \ldots, n\} \).

2. **Condition Analysis**:
   - We need to ensure that there do not exist positive integers \( x_1, x_2, y_1, y_2, y_3 \) such that \( x_1 < x_2 \), \( y_1 < y_2 < y_3 \), and the points \((x_1, y_1)\), \((x_1, y_2)\), \((x_1, y_3)\), \((x_2, y_2)\) are all in \( A \).

3. **Column Burning Concept**:
   - A column \( c \) is considered "burned" if there exists a row \( r \) and two additional columns \( c_1 \) and \( c_2 \) with \( c_1 < c < c_2 \) such that \((r, c_1)\), \((r, c)\), \((r, c_2)\) are all shaded. This means that any additional cell below this row in column \( c \) would complete a "T" shape, violating the condition.

4. **Upper Bound Calculation**:
   - For each row, ignore the leftmost and rightmost shaded cells. There are \( 2m \) such cells (since there are \( m \) rows and each row can have at most 2 such cells).
   - The remaining cells in each row can potentially burn their corresponding columns. Since there are \( n-2 \) columns that can be burned (excluding the first and last columns), the maximum number of such cells is \( n-2 \).

5. **Combining the Counts**:
   - The total number of cells is the sum of the leftmost and rightmost cells in each row plus the maximum number of cells that can burn columns:
   \[
   2m + (n-2)
   \]

6. **Constructing the Set \( A \)**:
   - To achieve this maximum, we can shade the entire leftmost column, the entire rightmost column, and the entire bottom row. This forms a "U" shape and ensures that no "T" shape is formed.

Thus, the largest possible number of elements in \( A \) is:
\[
\boxed{2m + n - 2}
\]