Let $ p $ be a natural number and let two partitions $ \mathcal{A} =\left\{ A_1,A_2,...,A_p\right\} ,\mathcal{B}=\left\{ B_1,B_2,...B_p\right\} $ of a finite set $ \mathcal{M} . $ Knowing that, whenever an element of $ \mathcal{A} $ doesn´t have any elements in common with another of $ \mathcal{B} , $ it holds that the number of elements of these two is greater than $ p, $ prove that $ \big| \mathcal{M}\big|\ge\frac{1}{2}\left( 1+p^2\right) . $ Can equality hold?
1. Let \( \mathcal{A} = \{A_1, A_2, \ldots, A_p\} \) and \( \mathcal{B} = \{B_1, B_2, \ldots, B_p\} \) be two partitions of a finite set \( \mathcal{M} \). Let \( a_i = |A_i| \) and \( b_i = |B_i| \) for \( i = 1, 2, \ldots, p \). Without loss of generality, assume \( a_i \) and \( b_i \) are in non-decreasing order.

2. Suppose, for the sake of contradiction, that \( |\mathcal{M}| < \frac{1 + p^2}{2} \).

3. Since \( \mathcal{A} \) and \( \mathcal{B} \) are partitions of \( \mathcal{M} \), we have:
   \[
   \sum_{i=1}^p a_i = |\mathcal{M}| \quad \text{and} \quad \sum_{i=1}^p b_i = |\mathcal{M}|
   \]

4. Given \( |\mathcal{M}| < \frac{1 + p^2}{2} \), we have:
   \[
   a_1 < \frac{1 + p^2}{2p}
   \]

5. Since \( a_1 \) is the smallest element in the partition \( \mathcal{A} \), there are at least \( \lceil p - \frac{1 + p^2}{2p} \rceil \) sets from \( \mathcal{B} \) that do not have common elements with \( A_1 \).

6. The number of elements in \( A_1 \) and any \( B_j \) that does not intersect with \( A_1 \) must be greater than \( p \). Therefore, we have:
   \[
   |A_1| + |B_j| > p
   \]

7. Since there are at least \( \lceil p - \frac{1 + p^2}{2p} \rceil \) such sets \( B_j \), we get:
   \[
   |\mathcal{M}| \geq (p + 1) \left\lceil p - \frac{1 + p^2}{2p} \right\rceil
   \]

8. Simplifying the expression inside the ceiling function:
   \[
   p - \frac{1 + p^2}{2p} = \frac{2p^2 - (1 + p^2)}{2p} = \frac{p^2 - 1}{2p}
   \]

9. Therefore:
   \[
   \left\lceil \frac{p^2 - 1}{2p} \right\rceil \geq \frac{p^2 - 1}{2p}
   \]

10. Thus:
    \[
    |\mathcal{M}| \geq (p + 1) \left\lceil \frac{p^2 - 1}{2p} \right\rceil \geq (p + 1) \frac{p^2 - 1}{2p}
    \]

11. Simplifying further:
    \[
    (p + 1) \frac{p^2 - 1}{2p} = \frac{(p + 1)(p^2 - 1)}{2p} = \frac{p^3 + p^2 - p - 1}{2p}
    \]

12. For \( p = 2k \) or \( p = 2k + 1 \), we need to show that:
    \[
    \frac{p^3 + p^2 - p - 1}{2p} \geq \frac{1 + p^2}{2}
    \]

13. Simplifying the inequality:
    \[
    \frac{p^3 + p^2 - p - 1}{2p} \geq \frac{1 + p^2}{2}
    \]

14. Multiplying both sides by \( 2p \):
    \[
    p^3 + p^2 - p - 1 \geq p^2 + p
    \]

15. Simplifying:
    \[
    p^3 - 2p \geq 1
    \]

16. For \( p \geq 2 \), the inequality holds true, leading to a contradiction.

17. Therefore, our initial assumption that \( |\mathcal{M}| < \frac{1 + p^2}{2} \) must be false. Hence, we have:
    \[
    |\mathcal{M}| \geq \frac{1 + p^2}{2}
    \]

18. To check if equality can hold, consider the case when \( |\mathcal{M}| = \frac{1 + p^2}{2} \). If \( p \) is even, say \( p = 2k \), then \( |\mathcal{M}| = \frac{1 + (2k)^2}{2} = 2k^2 + \frac{1}{2} \), which is not an integer. If \( p \) is odd, say \( p = 2k + 1 \), then \( |\mathcal{M}| = \frac{1 + (2k + 1)^2}{2} = 2k^2 + 2k + 1 \), which is an integer. Therefore, equality can hold when \( p \) is odd.

The final answer is \( \boxed{ |\mathcal{M}| \geq \frac{1 + p^2}{2} } \). Equality can hold when \( p \) is odd.