Bounding, splitting, and almost disjointness

Splitting, Bounding, and Almost Disjointness Can Be Quite Different

We prove the consistency of add(N ) < cov(N ) < p = s < add(M) = cof(M) < a = non(N ) = c with ZFC where each of these cardinal invariants assume arbitrary uncountable regular

Razborov Disjointness Lower Bound , Forster ’ S Theorem

In this lecture, we show two results dealing with lower bounds in communication complexity. The first lower bound is an Ω(n) lower bound on the distributional complexity of Disjointness due to [3, 8]. Here we will present the simplified proof presented in [8]. In the second part, we will show how to obtain lower bounds on the unbounded error probabilistic communication complexity by Forster’s m...

Splittings of Groups and Intersection Numbers

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple closed curve, and that two simple closed curves on a surface with intersection number zero can be isotoped to be disjoint. In [15] and [16], analogues of the classical JSJ-decompositions of Haken manifolds were described using the notion of isotopic disjoint...

Communication Complexity Theory: Thirty-Five Years of Set Disjointness

The set disjointness problem features k communicating parties and k subsets S1, S2, . . . , Sk ⊆ {1, 2, . . . , n}. No single party knows all k subsets, and the objective is to determine with minimal communication whether the k subsets have nonempty intersection. The important special case k = 2 corresponds to two parties trying to determine whether their respective sets intersect. The study of...

The Construction of Balanced Bounding-Volume Hierarchies using Spatial Object Median Splitting Method for Collision Detection