Name four border slave states that remained loyal to the union?

Delaware remained loyal to the Union. Maryland might have seceded had Lincoln not placed Federal troops around the state legislature, but the state remained in the Union. Kentucky did not secede but could not entirely be said to be loyal to the Union. It was officially neutral. About twice as many regiments from Kentucky joined the Confederacy as joined the Union, but it is likely that some Kentucky men joined with regiments from other states both north and south. Missouri was a slave state and never formally seceded. Although support for the South was strong, and a pro-Confederacy Army and legislature formed, the state remained in the Union. Estimates of the time say that Missouri was 90% for the Confederacy and 10% Union.

The last one to name here was West Virginia, which was part of Virginia until several counties banded together to secede from Virginia to rejoin the Union as a new state in 1863. West Virginia never existed as a slave state, having been brought into the Union after the Emancipation Proclamation, and is therefore a tricky but incorrect answer to this question.