I was wondering if there exists an $\alpha>0$ and a $d$-regular bipartite$(L \cup R)$ graph with $2n$ vertices such that any subset $S$ of size at most $\alpha n$ in $L$ will have at least $(d-1.01)|S|$ neighbors in $R$.
There is a result in https://people.seas.harvard.edu/~salil/pseudorandomness/expanders.pdf
For all constants $D \geq 3$, there is a constant $\alpha>0$ such that for all $N$, a random $D$-regular undirected graph on $N$ vertices is an $(αN,D −1.01)$ vertex expander with probability at least $1/2$. I was wondering if the double cover of such a graph gives the result I want?