We shall show that the union-closed families conjecture holds for a random union-closed family with high probability. This strengthens a recent result of Bruhn and Schaudt.

We prove that, with high probability, the space complexity of refuting a random unsatisfiable boolean formula in -CNF on variables and clauses is .

The complete proof is to be found in Vieille [4],[5]. The general idea of the proof is to deÞne a class of sets, the solvable sets, which can safely be thought of as absorbing states, and to construct ε-equilibrium proÞles such that the induced play reaches one of these sets with high probability, and remains there. Solvable sets are deÞned in Section 1. The organization of the proof is present...

We show that with high probability a random subset of {1, . . . , n} of size Θ(n) contains two elements a and a + d, where d is a positive integer. As a consequence, we prove an analogue of the Sárközy-Fürstenberg theorem for a random subset of {1, . . . , n}.

We explore further the hardness of the non-dual discrete variant of the Ring-LWE problem for various number rings, give improved attacks for certain rings satisfying some additional assumptions, construct a new family of vulnerable Galois number fields, and apply some number theoretic results on Gauss sums to deduce the likely failure of these attacks for 2-power cyclotomic rings and unramified...