In what follows, I will have most variables ranging over positive integers 
(or sets of positive integers,
or even sets of sets of positive integers). Let $n \gt 1$, and consider an interval $I$
 of $n$ consecutive integers $[a+1,\ldots, a+n]$.  Consider the subset $L$ (depending on $I$)
of $P(I)$ of $I$ intersected with maximal antichains in 
 the integer divisibility poset (actually quasi order, but most of the time
will be spent in the positive integer part, which looks like a lattice; $0 \lt -a \lt n$ may be considered later),
 so $M \in L$ iff 1) for all $x,y \in M$, either $x=y$ or
$\gcd(x,y)=1$  and 2) for all $z \in  I - M$ there is $x \in M$ with $\gcd(x,z) \gt 1$ .

Since any two consecutive positive integers are coprime, one has $\card(M) \ge 2$.  If $d$ is a
multiple of $\pi(n)$ primorial and $d$ happens to be in $M$, then $\card(M) \lt 4$.  However, in
this same interval containing $d$, we can choose a set $N$ that "looks like"
${d+1, d+2, \ldots, d+p_k}$ where $k$ is $O(\pi(n))$ and $p_j$ is the $j$th (positive) prime.  Based
on this example, I am confident (but can not yet prove)
that a lower bound for the maximum of the cardinalities of sets in $L$  is $\pi(n/2) + 1$.

Now let $I_t =\{ m \in I, m $is an integer multiple of $t\}$
For any $M \in L$, we must have $\card(M \cap I_t) \lt 2$ for any prime $t$.  So an
upper bound for $\card(M)$ is $\pi(n) + \rho(n)$, where $\rho(n)$ is the largest number of
integers relatively prime to $P_n$ (the $n$th primorial) in any subset of shape $I$
(collection of $n$ consecutive integers).  

I do not have a good expression for $\rho(n)/n$, but it is related to the product
$\prod_{i \le n} (1 - 1/p_i)$.  I am trying to bound this product from below by $1/2\ln(n\ln(n))$,
but there are some recent oscillation results by Diamond and Pintz that make
me unsure when the bound actually holds. It is related to the MathOverflow question
http://mathoverflow.net/questions/37679/erik-westzynthiuss-cool-upper-bound-argument-update
which I will update soon (but with results modulo oscillation, rather than absolute results).

Gerhard "Ask Me About System Design" Paseman, 2011.02.20