Suppose that $10$ points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.
1. **Restate the problem and initial assumptions:**
   We are given 10 points in the plane such that among any five of them, there are four that lie on a circle. We need to find the minimum number of these points that must lie on a single circle.

2. **Generalize the problem for \( n \geq 5 \):**
   If we replace 10 with \( n \geq 5 \), we hypothesize that the answer is \( n-1 \) for \( n \geq 7 \) and 4 for \( n = 5, 6 \).

3. **Verify the cases for \( n = 5 \) and \( n = 6 \):**
   - For \( n = 5 \), it is obvious that if any four out of five points lie on a circle, then all five points must lie on the same circle.
   - For \( n = 6 \), consider the six vertices of two unit squares in the plane that share a side. Any five points chosen from these six will have four points that lie on a circle.

4. **Assume \( n \geq 7 \) and find a circle with 5 points:**
   - Suppose we can find a circle with at least five points, say \( A, B, C, D, E \), and two other points \( X \) and \( Y \) not on that circle.
   - Consider the group \( X, Y, A, B, C \). Since four of these must be concyclic, one of \( A, B, C \) must not belong to this group. Assume WLOG that the group is \( X, Y, A, B \).

5. **Analyze the groups to find contradictions:**
   - Look at the group \( X, Y, A, C, D \). The four concyclic points must include \( X \) and \( Y \), so it cannot include \( A \). Thus, the group is \( X, Y, C, D \).
   - Now consider the group \( X, Y, A, C, E \). The four concyclic points must include \( X \) and \( Y \), but cannot include \( A \) or \( C \), leading to a contradiction.

6. **Show that no \( n \geq 7 \) works:**
   - To prove this, it suffices to show it for \( n = 7 \).
   - Let \( A, B, C, D \) be four concyclic points. Let \( X, Y, Z \) be the other three points.
   - Consider circles through the quadruples \( (X, Y, A, B) \), \( (X, Y, C, D) \), \( (X, Z, A, C) \), and \( (X, Z, B, D) \).

7. **Perform an inversion with pole \( X \):**
   - For a general point \( P \), denote its image by \( P' \).
   - Points \( A', B', C', D' \) are still concyclic, and \( Y' = A'B' \cap C'D' \), \( Z' = A'C' \cap B'D' \).
   - The six points \( A', B', C', D', Y', Z' \) must still satisfy the hypothesis, but no four of \( Y', Z', A', C', D' \) are concyclic, leading to a contradiction.

Thus, the minimum number of points that must lie on a circle is \( 9 \).

The final answer is \( \boxed{9} \).