We prove that a small-depth Frege refutation of the Tseitin contradiction
on the grid requires subexponential size.
We conclude that polynomial size Frege refutations
of the Tseitin contradiction must use formulas of almost
logarithmic depth.

Some more details added and some arguments expanded.

We prove that a small-depth Frege refutation of the Tseitin contradiction
on the grid requires subexponential size.
We conclude that polynomial size Frege refutations
of the Tseitin contradiction must use formulas of almost
logarithmic depth.