elusive Josef Farkas

I have no idea where in Hungary / Slovakia my great grandfather came from. Maybe someone out there will connect these facts to a story they have heard....

Here is what I know

Joseph was born about 1853 (He may have had a middle initial "A" he may also have been known as Peter Josef)
Married Maria around 1881
Josef came to america alone in 1882
Maria came to America about 1896. She may have had 2 children born in Hungary, Frank and Ignatz (I know for sure their was a Frank, my mother met his children)

Maria and Joseph had 3 children born in America
Clara, Julius, Anna

While in America, without his wife, Josef had 4 sons born to a woman named Erszi Vasko
John, Louis, James, Josef