We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials over F 2 . We prove that almost all degree d polynomials have only an exponentially small correlation with all polynomials of degree at most d − 1, for all degrees d up to Θ (n). That is, a random degree d polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low-degree polynomial. Recently, several results regarding the weight distribution of Reed-Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed-Muller codes.

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing?We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.

Abstract:Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k-uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges. C 2013 Wiley Periodicals, Inc. J. Combin. Designs 22: [488][489][490][491][492][493][494] 2014

Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists an (n, k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2,. .. , n} to r bits. Monotone encodings are relevant to the study of tamperproof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multiuser tracing ((k, α)-FUT families). We show that r(n, k) = Θ(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n, k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an additive constant.

Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists an (n, k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2, . . . , n} to r bits. Monotone encodings are relevant to the study of tamperproof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multi-user tracing ((k, α)-FUT families). We show that r(n, k) = Θ(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n, k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an additive constant.