Related papers: Which DNR can be minimal

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, decide whether there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain…

We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $\mu n$ for arbitrarily small $\mu>0$. About 20 years ago Koml\'os, Sark\"ozy, and Szemer\'edi resolved…

We explore the low levels of the structure of the continuous Weihrauch degrees of first-order problems. In particular, we show that there exists a minimal discontinuous first-order degree, namely that of $\accn$, without any determinacy…

For $0\le k\le n$, write $\binom nk=uv$ where the primes dividing $u$ are at most $k$ and the primes dividing $v$ exceed $k$, and let $f(n)$ be the least $k$ with $u>n^2$; Erd\H{o}s problem 684 asks for bounds on $f(n)$. We resolve the…

We prove that the smallest minimizer s(f) of a real convex function f is less than or equal to a real point x if and only if the right derivative of f at x is non-negative. Similarly, the largest minimizer t(f) is greater or equal to x if…

Let $\RR_S$ denote the expansion of the real ordered field by a family of real-valued functions $S$, where each function in $S$ is defined on a compact box and is a member of some quasianalytic class which is closed under the operations of…

In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The $n$-dimensional…

In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous…

The aim of this paper is twofold. First, we prove $L^p$ estimates for a regularized Green's function in three dimensions. We then establish new estimates for the discrete Green's function and obtain some positivity results. In particular,…

In this paper we study Weihrauch reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice…

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…

Let dec be the least cardinal kappa such that every function of first Baire class can be decomposed into kappa continuous functions. Cichon, Morayne, Pawlikowski and Solecki proved that cov(Meager) <= dec <= d and asked whether these…

The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size…

Let $\mathcal H$ be a reproducing kernel Hilbert space with a normalized complete Nevanlinna-Pick (CNP) kernel. We prove that if $(f_n)$ is a sequence of functions in $\mathcal H$ with $\sum\|f_n\|^2<\infty$, then there exists a contractive…

The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope of their minimal (with respect to pointwise…

We prove that the extremum stack of a discrete sequence is a minimal sufficient statistic for the class of all computable, causal, rate-independent functionals, in the sense of Kolmogorov complexity. Specifically, we establish K(Pi_n) -…

In this paper we study expander graphs and their minors. Specifically, we attempt to answer the following question: what is the largest function $f(n,\alpha,d)$, such that every $n$-vertex $\alpha$-expander with maximum vertex degree at…

We prove that for any decision tree calculating a boolean function $f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n \delta_i \Inf_i(f), \] where $\delta_i$ is the probability that the $i$th input variable is read and $\Inf_i(f)$ is…

We show that every $H$-minor-free graph that also excludes a $(k \times k)$-grid as a minor has treewidth/branchwidth bounded from above by a function $f(t,k)$ that is linear in $k$ and polynomial in $t := |V(H)|$. Such a result was proven…

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. A well-known conjecture of Erd\H{o}s and Hajnal states that for any fixed…