Let $G$ be a graph and $\alpha(G)$ the size of its largest independent vertex set. I have a partition of $V(G)$ into several subsets, each an independent vertex set of size less than $\alpha(G)$. My question is that, if I try all sorts of union of those subsets, am I bound to find a subset of $V(G)$ of size $\alpha(G)$?
Also, although this is a question on graph theory, I have the feeling that this has a lot to do with set theory (which I've never had a chance to study properly). Could anyone tell me if the following reformulation is correct?
Let $S$ be a set and $\alpha$ a property that a subset of $S$ might have. We know that the largest subset of $S$ with the property $\alpha$ has size $\alpha(S)$. Given a partition of $S$ into subsets with property $\alpha$, each with size less than $\alpha(S)$, can we find a collection of some of those subsets, whose size is $\alpha(S)$?
Take a graph with 4 nodes and only one edge, that connects nodes $1$ and $2$.
3 is the maximum number of independent nodes, but you can partition $V$ into two sets of independent vertices $\{1,3\}$ and $\{2,4\}$.
The only possible union has 4 elements, and is not composed by independent nodes.