  We discuss the problem of existence of latin squares without a substructure
consisting of six elements $(r_1,c_2,l_3)$, $(r_2,c_3,l_1)$, $(r_3,c_1,l_2)$,
$(r_2,c_1,l_3)$, $(r_3,c_2,l_1)$, $(r_1,c_3,l_2)$. Equivalently, the
corresponding latin square graph does not have an induced subgraph isomorphic
to $K_{3,3}$. The exhaustive search [Brouwer, Wanless. Universally
noncommutative loops. 2011] says that there are no such latin squares of order
from $3$ to $11$, and there are only two $K_{3,3}$-free latin squares of order
$8$, up to equivalence. We repeat the search, establishing also the number of
latin $m$-by-$n$ rectangles for each $m$ and $n$ less or equal to $11$. As a
switched combination of two orthogonal latin squares of order $8$, we construct
a $K_{3,3}$-free (universally noncommutative) latin square of order $16$.
  Keywords: latin square; transversal; trade; pattern avoiding; eigenfunction;
universally noncommutative loops.
