arXiv subjects
Publications and source records attributed to Jonathan Tidor.
Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.
Fix a prime $p$ and a positive integer $R$. We study the property testing of functions $\mathbb F_p^n\to[R]$. We say that a property is testable if there exists an oblivious tester for this property with one-sided error and constant query complexity. Furthermore, a property is proximity oblivious-testable (PO-testable) if the test is also independent of the proximity parameter $ε$. It is known that a number of natural properties such as linearity and being a low degree polynomial are PO-testable. These properties are examples of linear-invariant properties, meaning that they are preserved under linear automorphisms of the domain. Following work of Kaufman and Sudan, the study of linear-invariant properties has been an important problem in arithmetic property testing. A central conjecture in this field, proposed by Bhattacharyya, Grigorescu, and Shapira, is that a linear-invariant property is testable if and only if it is semi subspace-hereditary. We prove two results, the first resolves this conjecture and the second classifies PO-testable properties. (1) A linear-invariant property is testable if and only if it is semi subspace-hereditary. (2) A linear-invariant property is PO-testable if and only if it is locally characterized. Our innovations are two-fold. We give a more powerful version of the compactness argument first introduced by Alon and Shapira. This relies on a new strong arithmetic regularity lemma in which one mixes different levels of Gowers uniformity. This allows us to extend the work of Bhattacharyya, Fischer, Hatami, Hatami, and Lovett by removing the bounded complexity restriction in their work. Our second innovation is a novel recoloring technique called patching. This Ramsey-theoretic technique is critical for working in the linear-invariant setting and allows us to remove the translation-invariant restriction present in previous work.
We prove an arithmetic analog of the induced graph removal lemma for complexity 1 patterns over finite fields. Informally speaking, we show that given a fixed collection of $r$-colored complexity 1 arithmetic patterns over $\mathbb F_q$, every coloring $ϕ\colon \mathbb F_q^n \setminus\{0\} \to [r]$ with $o(1)$ density of every such pattern can be recolored on an $o(1)$-fraction of the space so that no such pattern remains.
In the framework of graph property testing, we study the problem of determining if a graph admits a cluster structure. We say that a graph is $(k, ϕ)$-clusterable if it can be partitioned into at most $k$ parts such that each part has conductance at least $ϕ$. We present an algorithm that accepts all graphs that are $(2, ϕ)$-clusterable with probability at least $\frac{2}3$ and rejects all graphs that are $ε$-far from $(2, ϕ^*)$-clusterable for $ϕ^* \le μϕ^2 ε^2$ with probability at least $\frac{2}3$ where $μ> 0$ is a parameter that affects the query complexity. This improves upon the work of Czumaj, Peng, and Sohler by removing a $\log n$ factor from the denominator of the bound on $ϕ^*$ for the case of $k=2$. Our work was concurrent with the work of Chiplunkar et al.\@ who achieved the same improvement for all values of $k$. Our approach for the case $k=2$ relies on the geometric structure of the eigenvectors of the graph Laplacian and results in an algorithm with query complexity $O(n^{1/2+O(1)μ} \cdot \text{poly}(1/ε, 1/ϕ,\log n))$.
The critical threshold of a (simple binary) matroid $N$ is the infimum over all $ρ$ such that any $N$-free matroid $M$ with $|M|>\rho2^{r(M)}$ has bounded critical number. In this paper, we resolve two conjectures of Geelen and Nelson, showing that the critical threshold of the projective geometry $PG(t-1,2)$ is $1-3\cdot2^{-t}$. We do so by proving the following stronger statement: if $M$ is $PG(t-1,2)$-free with $|M|>(1-3\cdot2^{-t})2^{r(M)}$, then the critical number of $M$ is $t-1$ or $t$. Together with earlier results of Geelen and Nelson [GN14] and Govaerts and Storme [GS06], this completes the classification of dense $PG(t-1,2)$-free matroids.
Consider a set $A$ with no $p$-term arithmetic progressions for $p$ prime. The $p$-Stanley sequence of a set $A$ is generated by greedily adding successive integers that do not create a $p$-term arithmetic progression. For $p>3$ prime, we give two distinct constructions for $p$-Stanley sequences which have a regular structure and satisfy certain conditions in order to be modular $p$-Stanley sequences, a set of particularly nice sequences defined by Moy and Rolnick which always have a regular structure. Odlyzko and Stanley conjectured that the 3-Stanley sequence generated by $\{0,n\}$ only has a regular structure if $n=3^k$ or $n=2\cdot 3^k$. For $p>3$ we find a substantially larger class of integers $n$ such that the $p$-Stanley sequence generated from $\{0,n\}$ is a modular $p$-Stanley sequence and numerical evidence given by Moy and Rolnick suggests that these are the only $n$ for which the $p$-Stanley sequence generated by $\{0,n\}$ is a modular $p$-Stanley sequence. Our second class is a generalization of a construction of Rolnick for $p=3$ and is thematically similar to the analogous construction by Rolnick.
We discuss two approaches to a recent question of Loh: must a 3-colored transitive tournament on $N$ vertices have a 1-color-\emph{avoiding} path of vertex-length at least $N^{2/3}$? This question generalizes the Erdős--Szekeres theorem on monotone subsequences. First, we define three canonical transformations on these tournaments called Color, Record, and Dual. We use these to establish a reduction to special tournaments with natural geometric and combinatorial properties. In many cases (including all known tight examples), these tournaments have recursive Gallai decompositions. Not all relevant tournaments have Gallai decompositions, but those that do satisfy the desired $N^{2/3}$ bound by recent work of Wagner, roughly analogous to earlier work of Fox, Grinshpun, and Pach on a similar \emph{undirected} problem. Second, we consider the related geometric problem of bounding \emph{slice-increasing} sets $S\subseteq [n]^3$, which---under an additional ordering hypothesis on $S$---was shown by Loh to be equivalent to the original question. In particular, we establish a rigorous connection from a problem of Szabó and Tardos, raise a stronger $L^2$-question on slice-counts, and mention a surprising overlap with the joints problem.
An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters. If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be the maximum length of any $r$-sparse sequence with $n$ distinct letters which has no subsequence isomorphic to $u$. For every sequence $u$ define $\mathit{fw}(u)$, the formation width of $u$, to be the minimum $s$ for which there exists $r$ such that there is a subsequence isomorphic to $u$ in every $(r, s)$-formation. We use $\mathit{fw}(u)$ to prove upper bounds on $\mathit{Ex}(u, n)$ for sequences $u$ such that $u$ contains an alternation with the same formation width as $u$. We generalize Nivasch's bounds on $\mathit{Ex}((ab)^{t}, n)$ by showing that $\mathit{fw}((12 \ldots l)^{t})=2t-1$ and $\mathit{Ex}((12\ldots l)^{t}, n) =n2^{\frac{1}{(t-2)!}α(n)^{t-2}\pm O(α(n)^{t-3})}$ for every $l \geq 2$ and $t\geq 3$, such that $α(n)$ denotes the inverse Ackermann function. Upper bounds on $\mathit{Ex}((12 \ldots l)^{t} , n)$ have been used in other papers to bound the maximum number of edges in $k$-quasiplanar graphs on $n$ vertices with no pair of edges intersecting in more than $O(1)$ points. If $u$ is any sequence of the form $a v a v' a$ such that $a$ is a letter, $v$ is a nonempty sequence excluding $a$ with no repeated letters and $v'$ is obtained from $v$ by only moving the first letter of $v$ to another place in $v$, then we show that $\mathit{fw}(u)=4$ and $\mathit{Ex}(u, n) =Θ(nα(n))$. Furthermore we prove that $\mathit{fw}(abc(acb)^{t})=2t+1$ and $\mathit{Ex}(abc(acb)^{t}, n) = n2^{\frac{1}{(t-1)!}α(n)^{t-1}\pm O(α(n)^{t-2})}$ for every $t\geq 2$.
We examine semi-bar visibility graphs in the plane and on a cylinder in which sightlines can pass through k objects. We show every semi-bar k-visibility graph has a (k+2)-quasiplanar representation in the plane with vertices drawn as points in convex position and edges drawn as segments. We also show that the graphs having cylindrical semi-bar k-visibility representations with semi-bars of different lengths are the same as the (2k+2)-degenerate graphs having edge-maximal (k+2)-quasiplanar representations in the plane with vertices drawn as points in convex position and edges drawn as segments.