# Unique-neighbor expander

I want to solve Problem 4.10 from Randomness by Salil Vadhan. https://people.seas.harvard.edu/~salil/cs225/spring15/PS3.pdf

Consider a bipartite expander $$G$$ with left degree $$D$$ so that every subset $$S$$ of the left vertices with at most $$K$$ vertices has at least $$(1-\epsilon)D|S|$$ neighbors. Then $$G$$ also has the property that it has $$(1-2\epsilon)D|S|$$ unique neighbors. Unique meaning that it has exactly one corresponding vertex from $$S$$.

I recognize that the new expansion factor is $$(1-2\epsilon)D = 2\cdot(1-\epsilon)D -D$$