Choose the correct form of the word to complete the sentence: Their/there is only one house on the block with a unite picket fence in the front yard.

The correct choice is "there." In this context, "there" is used to indicate the existence of something, specifically referring to the one house on the block. It is an adverb that helps to describe location or existence.

Using "there" in the sentence communicates that one house is present, which aligns with the intended meaning. The other options do not fit: "their" is a possessive pronoun used to show ownership, "they're" is a contraction for "they are," and "thier" is a misspelling of "their." Each of these words serves a different grammatical function and would not appropriately convey the intended meaning of existence in this context.