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Register a new userA universal exponent for homeomorphs
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We prove a uniform bound on the topological Turan number of an arbitrary two-dimensional simplicial complex $S$: any $n$-vertex two-dimensional complex with at least $C_S n^{3-1/5}$ facets contains a homeomorphic copy of $S$, where $C_S > 0$ is an absolute constant depending on $S$ alone. This result, a two-dimensional analogue of a classical result of Mader for one-dimensional complexes, sheds some light on an old problem of Linial from 2006.
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Our first main result is a uniform bound, in every dimension $k in mathbb N$, on the topological Turan numbers of $k$-dimensional simplicial complexes: for each $k in mathbb N$, there is a $lambda_k ge k^{-2k^2}$ such that for any $k$-complex $mathcal{S}$, every $k$-complex on $n ge n_0(mathcal{S})$ vertices with at least $n^{k+1 - lambda_k}$ facets contains a homeomorphic copy of $mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $lambda_1$ is a result of Mader from 1967, and the existence of $lambda_2$ was suggested by Linial in 2006 and recently proved by Keevash-Long-Narayanan-Scott. We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, dots, V_r$ is said to be $d$-trace-bounded if for each $2 le i le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 cup V_2 cup dots cup V_i$. Our second main result is the following estimate for the Turan numbers of degenerate trace-bounded hypergraphs: for all $r ge 2$ and $dinmathbb N$, there is an $alpha_{r,d} ge (5rd)^{1-r}$ such that for any $d$-trace-bounded $r$-partite $r$-graph $H$, every $r$-graph on $n ge n_0(H)$ vertices with at least $n^{r - alpha_{r,d}}$ edges contains a copy of $H$. This strengthens a result of Conlon-Fox-Sudakov from 2009 who showed that such a bound holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).
A surprising result of FitzGerald and Horn (1977) shows that $A^{circ alpha} := (a_{ij}^alpha)$ is positive semidefinite (p.s.d.) for every entrywise nonnegative $n times n$ p.s.d. matrix $A = (a_{ij})$ if and only if $alpha$ is a positive integer or $alpha geq n-2$. Given a graph $G$, we consider the refined problem of characterizing the set $mathcal{H}_G$ of entrywise powers preserving positivity for matrices with a zero pattern encoded by $G$. Using algebraic and combinatorial methods, we study how the geometry of $G$ influences the set $mathcal{H}_G$. Our treatment provides new and exciting connections between combinatorics and analysis, and leads us to introduce and compute a new graph invariant called the critical exponent.
A universal cycle, or u-cycle, for a given set of words is a circular word that contains each word from the set exactly once as a contiguous subword. The celebrated de Bruijn sequences are a particular case of such a u-cycle, where a set in question is the set $A^n$ of all words of length $n$ over a $k$-letter alphabet $A$. A universal word, or u-word, is a linear, i.e. non-circular, version of the notion of a u-cycle, and it is defined similarly. Removing some words in $A^n$ may, or may not, result in a set of words for which u-cycle, or u-word, exists. The goal of this paper is to study the probability of existence of the universal objects in such a situation. We give lower bounds for the probability in general cases, and also derive explicit answers for the case of removing up to two words in $A^n$, or the case when $k=2$ and $nleq 4$.
A universal cycle for permutations of length $n$ is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length $n$, and containing all permutations of length $n$ as factors. It is well known that universal cycles for permutations of length $n$ exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, we offer a simple way to generate a universal cycle for permutations of length $n$, which is based on applying a greedy algorithm to a permutation of length $n-1$. We prove that this approach gives a unique universal cycle $Pi_n$ for permutations, and we study properties of $Pi_n$.
This paper has two related parts. The first generalizes Hochsters formula on resolutions of Stanley-Reisner rings to a colorful version, applicable to any proper vertex-coloring of a simplicial complex. The second part examines a universal system of parameters for Stanley-Reisner rings of simplicial complexes, and more generally, face rings of simplicial posets. These parameters have good properties, including being fixed under symmetries, and detecting depth of the face ring. Moreover, when resolving the face ring over these parameters, the shape is predicted, conjecturally, by the colorful Hochster formula.
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