A complete characterization of Nash-solvability of bimatrix games in terms of the exclusion of certain $2\times 2$ subgames

has a saddle point whenever every
$2 \times 2$ submatrix of it has one.
In contrast, a bimatrix game may have no
Nash equilibrium (NE) even when every
$2 \times 2$ subgame of it has one.
Nevertheless, Shapley's claim can be
generalized for bimatrix games in many ways as follows.
We partition all $2 \times 2$ bimatrix games
into fifteen classes $C = \{c_1, \ldots, c_{15}\}$
depending on the preference pre-orders of the two players.
A subset $t \subseteq C$ is called a NE-theorem
if a bimatrix game has a NE whenever
it contains no subgame from $t$.
We suggest a general method for getting
all minimal (that is, strongest) NE-theorems
based on the procedure of joint generation
of transversal hypergraphs given by a special oracle.
By this method we obtain all (six) minimal NE-theorems.