Find, with proof, the minimum positive integer n with the following property: for
any coloring of the integers $\{1, 2, . . . , n\}$ using the colors red and blue (that is, assigning the
color “red” or “blue” to each integer in the set), there exist distinct integers a, b, c between
1 and n, inclusive, all of the same color, such that $2a + b = c.$
To find the minimum positive integer \( n \) such that for any coloring of the integers \(\{1, 2, \ldots, n\}\) using the colors red and blue, there exist distinct integers \(a, b, c\) all of the same color such that \(2a + b = c\), we will use the concept of Ramsey theory.

1. **Claim**: The minimum \( n \) is \( \boxed{15} \).

2. **Proof that \( n = 15 \) works**:
   - Consider any coloring of the integers \(\{1, 2, \ldots, 15\}\) using red and blue.
   - We need to show that there exist distinct integers \(a, b, c\) all of the same color such that \(2a + b = c\).

3. **Using the Pigeonhole Principle**:
   - If we take any 3 integers from our set, at least 2 of them must be the same color.
   - We will consider different cases based on the number of integers of each color.

4. **Case Analysis**:
   - **Case 1**: \(1, 2, 3\) are the same color.
     - WLOG, let \(1, 2, 3\) be blue.
     - We need to find \(a, b, c\) such that \(2a + b = c\).
     - If \(4\) or \(5\) is blue, then \( (a, b, c) = (1, 2, 4) \) or \( (2, 1, 5) \) works.
     - If \(4\) and \(5\) are red, then consider higher numbers. We can find solutions like \( (4, 5, 13) \) or \( (5, 4, 14) \).

   - **Case 2**: \(1, 2\) are the same color and \(3\) is a different color.
     - WLOG, let \(1, 2\) be blue and \(3\) be red.
     - We need to find \(a, b, c\) such that \(2a + b = c\).
     - If \(4\) or \(5\) is blue, then \( (a, b, c) = (1, 2, 4) \) or \( (2, 1, 5) \) works.
     - If \(4\) and \(5\) are red, then consider higher numbers. We can find solutions like \( (3, 4, 10) \) or \( (4, 3, 11) \).

   - **Case 3**: \(1, 3\) are the same color and \(2\) is a different color.
     - WLOG, let \(1, 3\) be blue and \(2\) be red.
     - We need to find \(a, b, c\) such that \(2a + b = c\).
     - If \(4\) or \(5\) is blue, then \( (a, b, c) = (1, 3, 7) \) or \( (3, 1, 7) \) works.
     - If \(4\) and \(5\) are red, then consider higher numbers. We can find solutions like \( (2, 4, 8) \) or \( (4, 2, 10) \).

5. **Counterexamples for \( n < 15 \)**:
   - For \( n = 14 \), consider the coloring: \(\{1, 2, 3, 13, 14\}\) blue and \(\{4, 5, 6, 7, 8, 9, 10, 11, 12\}\) red. There is no \(a, b, c\) all of the same color such that \(2a + b = c\).
   - Similar counterexamples can be constructed for \( n = 13, 12, \ldots, 1 \).

Therefore, the minimum \( n \) such that for any coloring of the integers \(\{1, 2, \ldots, n\}\) using red and blue, there exist distinct integers \(a, b, c\) all of the same color such that \(2a + b = c\) is \( \boxed{15} \).