Mantel’s theorem states that every n-vertex graph with $\lfloor \frac{n^2}{4}\rfloor +t$ edges, where $t>0$, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the Erdős–Rademacher problem. Lovász and Simonovits proved that there are at least $t\lfloor n∕2\rfloor$ triangles in each of those graphs. Katona and Xiao considered the same problem under the additional condition that there are no $s-1$ vertices covering all triangles. They settled the case $t=1$ and $s=2$. Solving their conjecture, we determine the minimum number of triangles for every fixed pair of s and t, when n is sufficiently large. Additionally, solving another conjecture of Katona and Xiao, we extend the theory for considering cliques instead of triangles.

In this paper, we present a new extension for Hardy–Littlewood-type inequalities using the Riemann–Liouville fractional integral with respect to another function in the weighted Lebesgue space $L_{\mathit{\psi }}^p[a,b]$. These estimates are very important in the study of fractional differential equations with variational form and in the construction of fractional spaces of Sobolev type.

We study the geometry of the Heisenberg group ${\mathrm{Nil}}_3$ with a balanced metric, the sum of the left and right invariant metrics. We prove that with this metric, ${\mathrm{Nil}}_3$ splits as a Riemannian product $\mathbb{T}\times \mathbb{Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of ${\mathrm{Nil}}_3$. So we prove the existence of complete properly embedded minimal surfaces in ${\mathrm{Nil}}_3$ by solving the asymptotic Dirichlet problem for the minimal surface equation on $\mathbb{T}$. We also show the existence of complete properly embedded minimal surfaces foliating an open set of ${\mathrm{Nil}}_3$ having as boundary a given curve Γ in $\mathbb{T}$, satisfying the exterior circle condition, by solving the exterior Dirichlet problem for the minimal surface equation in the unbounded connected component of $\mathbb{T}\setminus \mathrm{\Gamma }$.

We prove that if a metric space X has Nagata dimension zero with constant c, then there exists a dense subset of X that is $8c$-bi-Lipschitz equivalent to a weighted tree. The factor 8 is the best possible if $c=1$; that is, if X is an ultrametric space. This yields a new proof of a result of Chan, Xia, Konjevod, and Richa. Moreover, as an application, we also obtain quantitative versions of certain metric embedding and Lipschitz extension results of Lang and Schlichenmaier. Finally, we prove a variant of our main theorem for 0-hyperbolic proper metric spaces. This generalizes a result of Gupta.

In this paper, we extend a result of Chen regarding the solvability of the twisted deformed Hermitian Yang–Mills (dHYM) equations on compact Kähler manifolds to allow for the twisting function to be nonconstant and slightly negative in all dimensions. Using this result along with the methods of Datar and Pingali, we prove that the twisted dHYM equation on compact, projective manifolds can be solved provided certain numerical conditions are satisfied. As a corollary, we obtain a new proof in the projective case of a recent theorem of Chu, Lee, and Takahashi addressing a conjecture of Collins, Jacob, and Yau.

We introduce the notion of Burch submodules and weakly $\mathfrak{m}$-full submodules of modules over a local ring $(R,\mathfrak{m})$ and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test properties. We also show that over a local ring $(R,\mathfrak{m})$, a submodule M of a finitely generated R-module X, such that either $M=\mathfrak{m}X$ or $M(\subseteq \mathfrak{m}X$) is weakly $\mathfrak{m}$-full in X, is 1-Tor rigid, and a test module provided that X is faithful (and $X∕M$ has finite length when M is weakly $\mathfrak{m}$-full). As an application, we give some new class of modules and a new class of rings such that a conjecture of Huneke and Wiegand is affirmative for them.

Let $\mathrm{\pi }:P\to B$ be a smooth G-bundle over a compact Riemannian manifold B and c a smooth loop in B of constant seed $a(>0)$, where G is compact semi-simple Lie group. In this paper, we prove that the holonomy map ${\mathrm{hol}}_c:{\mathcal{A}}_P^{H^s}\to G$ is a homothetic submersion of coefficient a, where s is a nonnegative integer and ${\mathcal{A}}_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle P. In particular, we prove that if $s=0$, then ${\mathrm{hol}}_c$ has minimal regularizable fibers. From this fact, we can derive that each component of the inverse image of any equifocal submanifold in G by the holonomy map ${\mathrm{hol}}_c:{\mathcal{A}}_P^{H^0}\to G$ is an isoparametric submanifold in ${\mathcal{A}}_P^{H^0}$. As the result, we obtain a new systematic construction of isoparametric submanifolds in a Hilbert space.

The study of upper density problems on Ramsey theory was initiated by Erdős and Galvin in 1993 in the particular case of the infinite path, and by DeBiasio and McKenney in general. In this paper, we are concerned with the following problem: given a fixed finite graph F, what is the largest value of λ such that every 2-edge-coloring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite F-factor whose vertex set has upper density at least λ?

Here we prove a new lower bound for this problem. For some choices of F, including cliques and odd cycles, this new bound is sharp because it matches an older upper bound. For the particular case where F is a triangle, we also give an explicit lower bound of $1-\frac{1}{\sqrt{7}}=0.62203\dots$, improving the previous best bound of 3/5.

We explore two properties of backward orbits under semigroups of holomorphic self-maps in the unit disk. First, we prove that regular backward orbits are quasi-geodesics for the hyperbolic distance of the unit disk. Then we show that backward orbits satisfy a useful property, this time in Euclidean terms.