# Thread: How small can this contact network model be?

1. Hello All,

Suppose that we have (n) peers (or a company actual clients) denoted by (I) who are (pairwise) mutual friends as illustrated at the following: I1 - I2 - I3 - I4 - I5 - I6 - I7 - I8 - I9 -...- In - I1

Each (I or in fact actual client) is in relationship with a group of eight (8) 'acquaintances' (or potential clients), who are initially strangers for four (4) sequential peers/clients at the left and the right of that client. For example, acquaitances of peer/client (I5) are strangers for clients I1, I2, I3, and I4 at the left side and for clients I6, I7, I8, and I9 at the right.

As a result of this, a client's (i.e. I5) group of acquaintances must NOT have member in common with any group of acquaintances belonging to the four (4) sequential peers/clients at the left and the right sides of that client.
Another property of this contact network model is that NONE of any four (4) sequential clients must have mutual acquaintance with any other clients in that sequence. For example, acquaintances groups of I6, I7, I8, and I9 have NO member in common; in fact there are 32 distinctive acquaintances in sum for this sequence and 8 distinctive acquaintances for each client in that sequence.

The main purpose of this system is to know, with given (n) number of peers/clients, how many distinctive acquaintances (potential clients) AT LEAST is required for constructing this contact network without compromising on the properties/conditions described for the model.

Based on a preliminary research, I found Ramsey Theory may be of help; You may show me how I can apply Ramsey Theory for this or (if not applicable) guide me on how to solve this problem.

Best,
Faraz Davani