We extend results of Bonet, Buss and Pitassi on Bondy's Theorem and of Nozaki, Arai and Arai on Bollobas' Theorem by proving that Frankl's Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parameter t, we prove that Frankl's Theorem has polynomial size AC(0)-Frege proofs from instances of the pigeonhole principle.
Aisenberg, J., Bonet, M., Buss, S. Quasipolynomial size frege proofs of Frankl's Theorem on the trace of sets. "Journal of symbolic logic", 1 Juny 2016, vol. 81, núm. 2, p. 687-710.