  Two subgraphs $A,B$ of a graph $G$ are anticomplete if they are
vertex-disjoint and there are no edges joining them. Is it true that if $G$ is
a graph with bounded clique number, and sufficiently large chromatic number,
then it has two anticomplete subgraphs, both with large chromatic number? This
is a question raised by El-Zahar and Erd\H{o}s in 1986, and remains open. If
so, then at least there should be two anticomplete subgraphs both with large
minimum degree, and that is one of our results.
  We prove two variants of this. First, a strengthening: we can ask for one of
the two subgraphs to have large chromatic number: that is, for all $t, c\ge 1$
there exists $d\ge 1$ such that if $G$ has chromatic number at least $d$, and
does not contain the complete graph $K_t$ as a subgraph, then there are
anticomplete subgraphs $A,B$, where $A$ has minimum degree at least $c$ and $B$
has chromatic number at least $c$.
  Second, we look at what happens if we replace the hypothesis that $G$ has
sufficiently large chromatic number with the hypothesis that $G$ has
sufficently large minimum degree. This, together with excluding $K_t$, is {\em
not} enough to guarantee two anticomplete subgraphs both with large minimum
degree; but it works if instead of xcluding $K_t$ we exclude the complete
bipartite graph $K_{t,t}$. More exactly: for all $t, c\ge 1$ there exists $d\ge
1$ such that if $G$ has minimum degree at least $d$, and does not contain the
complete bipartite graph $K_{t,t}$ as a subgraph, then there are two
anticomplete subgraphs both with minimum degree at least $c$.
