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A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in R d given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a longstanding conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of "containing k colinear integer points."

S-gap shifts are an important and well-studied class of symbolic dynamical systems, and several generalizations of S-gap shifts have been proposed. This paper introduces a new class of symbolic dynamical systems called S-graph shifts which unites these generalizations with several other classes of symbolic dynamical systems using a graph-theoretic framework. The main result is a formula for the entropy of any S-graph shift, which, by specialization, resolves a problem raised by Matson and Sattler. We also show that every entropy value is obtained by uncountably many S-graph shifts and characterize those shifts that possess certain dynamical properties.

Gaussian filters have applications in a variety of areas in computer science from computer vision to speech recognition. The collapsing sum is a matrix operator that was recently introduced to study Gaussian filters combinatorially. In this paper, we determine the kernel of the collapsing sum and use it to derive a surprising isomorphism between additive groups of matrices. We also examine the recoverability of preimages as a matrix completion problem. Using bipartite graphs, we give an exact condition for when a partially-filled matrix can be extended to a preimage of a given matrix.