Barriers to show $P=NP$ We all know showing $P\ne NP$ has barriers. We all have studied these barriers because we believe $P\ne NP$.

However assume $P=NP$ and there are wise people who believe that possibility exists. If this is indeed the case then the very fact that we have not seen any good algorithms indicates there might be barriers in this alternate universe as well. Provability of $P\ne NP$ is barrier ridden and we do not know for sure $P\ne NP$ is the truth. We do not know for sure $P= NP$ is the truth either and so is provability of $P=NP$ also barrier ridden?
Mihalis Yannakakis has shown that the traveling salesman problem cannot be solved in polynomial time by using a symmetric linear program.

See the paper Expressing combinatorial optimization problems by Linear Programs, by Yannakakis.

This result was improved recently by  Fiorini,  Massar, Pokutta,  Tiwary, and De Wolf to drop the "symmetric" requirement in Yannakakis' result.