[Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem asks whether every finite lattice is the congruence lattice of a finite universal algebra. The answer is generally suspected to be no.

**Question 1**: What is the smallest lattice which is not known to be a congruence lattice of a finite universal algebra?

*Remark*: By "smallest" I mean "of smallest cardinal". By "not known" I don't mean that it should have been conjectured somewhere to be a counter-example.

[Pálfy and Pudlák 1980] proves that the above problem is equivalent to ask whether every finite lattice is the lattice of an interval in the subgroup lattice of a finite group. As pointed out by the authors, this theorem does not imply that every congruence lattice of a finite universal algebra is the lattice of an interval in the subgroup lattice of a finite group (whereas the converse is true), it is just an equivalence between two problems.

**Question 2**: What is the smallest lattice which is not known to be the lattice of an interval in the subgroup lattice of a finite group?

*Remark*: same as above.

This short course by William DeMeo provides a clear review of the background for this problem.