We consider Achlioptas processes for k-SAT formulas. We create a semi-random formula with n variables and m clauses, where each clause is a choice, made on-line, between two or more uniformly random clauses. Our goal is to delay the satisfiability/unsatisfiability transition, keeping the formula satisfiable up to densities m/n beyond the satisfiability threshold αk for random k-SAT. We show that three choices suffice to delay the transition for any k ≥ 3, and that two choices suffice for all 3 ≤ k ≤ 25. Wealso showthat two choices suffice tolower the threshold for all k ≥ 3, makingthe formula unsatisfiable at a density below αk. 1

We study two geometric models of random k-satisfiability which combine random k-SAT with the Random Geometric Graph: boolean literals are placed uniformly at random or according to a Poisson process in a cube, and for each set of k literals contained in a ball of a given radius, a clause is formed. For k = 2 we find the exact location of the satisfiability threshold (as either the radius or intensity of the Poisson process varies) and show the threshold is sharp; for k ≥ 3 we give bounds on the threshold that differ by a constant factor; and for one of the two models we prove that the threshold is in fact sharp for all k ≥ 2. 1

Abstract.There is stil much to discover about the mechanisms and nature of discontinuous percolation transitions. Much of the past work considers graph evolution algorithms known as Achlioptas processes in which a single edge is added to the graph from a set ofkrandomly chosen candidate edges at each timestep until a giant component emerges. Several Achlioptas processes seem to yield a discontinuous percolation transition, but it was proven by Riordan and Warnke that the transition must be continuous in the thermodynamic limit. However, they also proved that if the numberk(n) of candidate edges increases with the number of nodes, then the percolation transition may be discontinuous. Here we attempt to find the simplest such process which yields a discontinuous transition in the thermodynamic limit. We introduce a process which considers only the degree of candidate edges and not component size. We calculate the critical pointtc=(1−θ(1k))nand rigorously show that the critical window is of sizeO(nk(n)). Ifk(n)grows very slowly, for examplek(n)=logn, the critical window is barely sublinear and hence the phase transition is discontinuous but appears continuous in finite systems. We also present arguments that Achlioptas processes with bounded size rules wil always have continuous percolation transitions even with infinite choice. 1

Abstract Consider the random graph process where we start with an empty graph on n vertices, and at time t, are given an edge e t chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that whp the graph becomes Hamiltonian at time (1/2 + o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph has a directed Hamilton cycle whp only at time (1 + o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if at time t, instead of a random direction one is allowed to choose the orientation of e t , then whether it is possible or not to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that whp, the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.