For every epsilon and sufficiently large degree d, explicit d-regular graphs exist in which every small set S has at least (1-epsilon)d|S| distinct neighbors.
Explicit bounds for primes in arithmetic progressions
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Explicit Lossless Vertex Expanders
For every epsilon and sufficiently large degree d, explicit d-regular graphs exist in which every small set S has at least (1-epsilon)d|S| distinct neighbors.