Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition

Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces

Two applications of Nash-Williams' theory of barriers to sequences on Banach spaces are presented: The first one is the $c_0$-saturation of $C(K)$, $K$ countable compacta. The second one is the construction of weakly-null sequences generalizing the example of Maurey-Rosenthal

Small permutation classes

We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number $\kappa$, approximately 2.20557, for which there are only countably many permutation classes of growth rate (Stanley-Wilf limit) less than $\kappa$ but uncountably many permutation classes of growth rate $\kappa$, answering a question of Klazar. We go on to completely characterize the possible sub-$\kappa$ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial well-order, and atomicity (also known as the joint embedding property)

Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups

Let $W$ be an affine Weyl group, and let $\Bbbk$ be a field of characteristic $p>0$. The diagrammatic Hecke category $\mathcal{D}$ for $W$ over $\Bbbk$ is a categorification of the Hecke algebra for $W$ with rich connections to modular representation theory. We explicitly construct a functor from $\mathcal{D}$ to a matrix category which categorifies a recursive representation $\xi : \mathbb{Z}W \rightarrow M_{p^r}(\mathbb{Z}W)$, where $r$ is the rank of the underlying finite root system. This functor gives a method for understanding diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules which are "smaller" by a factor of $p$. It also explains the presence of self-similarity in the $p$-canonical basis, which has been observed in small examples. By decategorifying we obtain a new lower bound on the $p$-canonical basis, which corresponds to new lower bounds on the characters of the indecomposable tilting modules by the recent $p$-canonical tilting character formula due to Achar-Makisumi-Riche-Williamson.Comment: 62 pages, many figures, best viewed in colo

Two Approaches to Building Time-Windowed Geometric Data Structures

Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.\u27s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.\u27s O(n log^2 n) and O(n log n loglog n) solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems