A simple theory is a complete first-order theory with no tree property. A formula has the tree property when its parameters
can be arranged on an infinitely branching tree so that
the formulas along each infinite branch are jointly consistent,
but at each node any 
distinct immediate successors are inconsistent together, for some fixed finite 
.
Every stable theory is simple, but the converse fails. The theory of the random graph is a standard example of a simple unstable theory. Simple theories retain a useful independence relation given by nonforking. The stable forking conjecture asks whether their forking can always be witnessed by stable formulas.
