Related papers: A Tight Bound for Hypergraph Regularity II

In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with $k$-regular partitions. Extending the generating function for $k$-regular partitions…

A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general…

Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of…

Let $A_q(n,d)$ be the maximum order (maximum number of codewords) of a $q$-ary code of length $n$ and Hamming distance at least $d$. And let $A(n,d,w)$ that of a binary code of constant weight $w$. Building on results from algebraic graph…

In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…

We establish a strong, geometric lower bound on the (sequential) topological complexity of the unordered configuration spaces of a general graph. As an application, we show that, for most graphs, the topological complexity eventually…

We prove essentially sharp bounds for Ramsey numbers of ordered hypergraph matchings, inroduced recently by Dudek, Grytczuk, and Ruci\'{n}ski. Namely, for any $r \ge 2$ and $n \ge 2$, we show that any collection $\mathcal H$ of $n$ pairwise…

We give almost tight conditional lower bounds on the running time of the kHyperPath problem. Given an $r$-uniform hypergraph for some integer $r$, kHyperPath seeks a tight path of length $k$. That is, a sequence of $k$ nodes such that every…

In this paper, we prove a generalization of a conjecture of Erd\"{o}s, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph…

We establish sharp estimates for the convergence rate of the Kranosel'ski\v{\i}-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of…

We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a…

F.-H. Lin studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an…

Let T be a bounded linear operator acting on a complex Banach space X and (\lambda_n) a sequence of complex numbers. Our main result is that if |\lambda_n|/|\lambda_{n+1}| \to 1 and the sequence (\lambda_n T^n) is frequently universal then…

Erd\H{o}s and Lov\'asz noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $\tau(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We…

Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some…

For an $r$-uniform hypergraph $H$, let $\nu^{(m)}(H)$ denote the maximum size of a set~$M$ of edges in $H$ such that every two edges in $M$ intersect in less than $m$ vertices, and let $\tau^{(m)}(H)$ denote the minimum size of a collection…

The spectral Tur\'an theorem states that the $k$-partite Tur\'an graph is the unique graph attaining the maximum adjacency spectral radius among all graphs of order $n$ containing no the complete graph $K_{k+1}$ as a subgraph. This result…

We develop a connection between DP-colorings of $k$-uniform hypergraphs of order $n$ and coverings of $n$-dimensional Boolean hypercube by pairs of antipodal $(n-k)$-dimensional faces. Bernshteyn and Kostochka established that the lower…

We prove that for all $k \ge 3$ and any integers $\Delta, n$ with $n \ge 2^\Delta,$ there exists a $k$-graph on $n$ vertices with maximum degree at most $\Delta$ such that $r(H)\geq\tw_{k-1}(c_k \Delta) \cdot n$ for some constant $c_k > 0$,…

Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Tur\'an 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…