On smoothed analysis in dense graphs and formulas

We study a model of random graphs, where a random instance is obtained by adding random edges to a large graph of a given density. The research on this model has been started by Bohman and colleagues (Random Struct Algor 22 (2003), 33-42; Random Struct Algor 24 (2004), 105-117). Here we obtain a sharp threshold for the appearance of a fixed subgraph and for certain Ramsey properties. We also consider a related model of random &-SAT formulas, where an instance is obtained by adding random k-clauses to a fixed formula with a given number of clauses, and derive tight bounds for the non-satisfiability of the thus-obtained random formula.