Let me draw your attention to the following problem:

Consider a family of subsets of size d of the set N={1,2,…,n}.

Associate to a graph as follows: The vertices of are simply the sets in . Two vertices and are adjacent if .

For a subset let denote the subfamily of all subsets of which contain .

**MAIN ASSUMPTION**: Suppose that for every for which is not empty is **connected.**

**MAIN QUESTION: **How large can the diameter of be in terms of and .

If D(n, d) is the maximum diameter in terms of n and d, easy to see D(n, d) \geq D(n-1, d-1). Is there any easy inequalities that make D grow?

Dear Yosef,

Right. If you start with a family of (d-1)-subsets of a set of size n-1, and add a new element to the ground set and to every set in the family, you reach a new family with parameters n and d with precisely the same graph. I am not aware of general inequalities of this form which makes D grows substantially quicker than what follows from the inequality you have stated.

I was wondering if set-complements allow you to say that D(n, d) = D(n, n-d). I haven’t been able to show that the connected criterion must carry over to the complements in G(F’[A]), is it so?

The desired complementary behavior fails; consider F = { {1,2}, {2,3}, {3,4}, {4,5} }. Then we have:

F’ = { {1,2,3}, {1,2,5}, {1,4,5}, {3,4,5} }

Which does not supply a connected graph for A = {3}.

So far, I only have the trivial D(n,d) \geq D(n-k,d) + D(k,d) + d, for any choice of d \leq k \leq n-d, and d \geq 2.

This, plus D(n,d) \geq D(n-1,d-1) guarantees that D(n,d) \geq n-d . (This is not a particularly interesting revelation; this result can also be achieved by choosing F = { {1,2,…d}, {2,3,…,d+1}, …, {n-d+1,n-d+2,…,n} }.)

Yosef and Yosef’s Roomate

Indeed moving to complements is interesting. The connectivity condition is not preserved but is replaced by the condition that the induced family on every subset M of N, satisfies the condition that is connected. If we require connectivity

bothfor families and for families then the maximum diameter (for ) is .Has this been resolved? It seems like for d=2 and d=3, it is impossible to improve on the bound Yousef’s Roomate gave.

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