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Let p be a prime, and let L be a set of s congruence classes modulo p. Let $${\mathcal{H}}$$ be a family of subsets of [n] such that the size modulo p of each member of $${\mathcal{H}}$$ is not in L, but the size modulo p of every intersection of k distinct members of $${\mathcal{H}}$$ is in L. We prove that $$|{{\mathcal{H}}}|\le(k-1)\sum_{i=0}^s {n-1\choose i}$$ , improving the bound...
A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, we construct infinitely many one-regular graphs of valency 4 and 6. These graphs are Cayley graphs on dihedral groups.
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