Problem 1402

To prove that each messaging channel may be assigned one of 100 encryption keys such that no 4 users have the 6 pairwise channels between them all being assigned the same encryption key, we can proceed as follows:

1. Number the Users and Create Sets:
Number the users from $1$ to $105$. Create sets $I_1, I_2, \ldots, I_{53}$ where:
$$I_1 = \{1, 2\}, \quad I_2 = \{3, 4\}, \quad \ldots, \quad I_{52} = \{103, 104\}, \quad I_{53} = \{105\}$$
Each set $I_k$ contains 2 users, except for $I_{53}$ which contains 1 user.

2. Assign Encryption Keys:
For any two users $u$ and $v$, if $u \in I_k$ and $v \in I_\ell$ with $k \leq \ell$, assign their channel the key $k$.

3. Verify the Key Assignment:
Consider any four users. We need to show that the 6 pairwise channels between them do not all have the same encryption key.

- **Case 1: Four users are from three different sets $I_k, I_\ell, I_m$ with $k < \ell < m$:**
In this case, at least two different keys $k$ and $\ell$ are used among the channels. This is because the channels between users in different sets will have different keys.

- **Case 2: Four users are from two different sets $I_k$ and $I_\ell$:**
In this case, both keys $k$ and $\ell$ are used. For example, if two users are from $I_k$ and two users are from $I_\ell$, the channels between users in $I_k$ will have key $k$ and the channels between users in $I_\ell$ will have key $\ell$.

4. Conclusion:
In both cases, it is clear that not all 6 pairwise channels between any four users can have the same encryption key. Therefore, the assignment of keys as described ensures that no 4 users have all 6 pairwise channels between them assigned the same encryption key.

$\blacksquare$