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This article is part of my upcoming masters thesis which investigates thefollowing open problem from the book, Free Lattices, by R.Freese, J.Jezek, andJ.B. Nation published in 1995: "Which lattices (and in particular whichcountable lattices) are sublattices of a free lattice?"  Despite partial progress over the decades, the problem is still unsolved.There is emphasis on the countable case because the current body of knowledgeon sublattices of free lattices is most concentrated on when these sublatticesare countably infinite.  It is known that sublattices of free lattices which are finite can becharacterized as being those lattices which satisfy Whitman's condition and aresemidistributive. This assertion was conjectured by B. Jonsson in the 1960'sand proven by J.B. Nation in 1980. However, there is a desire for a new proofto this deep result as Nation's proof is very involved and more insight intosublattices of free lattices is sought after.  In this article, a sufficient condition involving a construct known as a joinminimal pair, or just a minimal pair, implying J'onsson's conjecture is proven.Minimal pairs were first defined by H. Gaskill when analysing sharplytransferable lattices. Using this sufficient condition, research by I.Rival andB.Sands is used to compare this condition with properties of finitesemidistributive lattices and in the process refute the main assertion of amanuscript by H.Muhle. Moreover, inspired by the approaches used by HenriMuhle, I will make a partial result which investigates a possible forbiddensublattice characterization involving breadth-two planar semidistributivelattices. To the best of my knowledge, the two results of this article (thesufficient condition for Jonsson's conjecture and the partial resultaforementioned) are new.

A Sufficient Condition for J\'onsson's Conjecture and its Relationship with Finite Semidistributive lattices