Related papers: Counterexamples to the B-spline conjecture for Gab…

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the…

We conjecture that the exceptional set in Manin's Conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this…

We investigate the combinatorial discrepancy of geometric set systems having bounded shallow cell complexity in the \emph{Beck-Fiala} setting, where each point belongs to at most $t$ ranges. For set systems with shallow cell complexity…

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph…

When $g\in\mathbb{N}$ we say that $A\subset\mathbb{N}$ is a $B_{2}[g]$ sequence if every $m\in\mathbb{N}$ has at most $g$ distinct representations of the shape $m=b_{1}+b_{2}$ with $b_{1}\leq b_{2}$ and $b_{1},b_{2}\in A$. We show for every…

Let $G$ be an abelian group. A set $A \subset G$ is a \emph{$B_k^+$-set} if whenever $a_1 + \dots + a_k = b_1 + \dots + b_k$ with $a_i, b_j \in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a…

Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is on a path…

The "No Gap Conjecture" of Br\"ustle-Dupont-P\'erotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two…

We prove a number field analogue of W. M. Schmidt's conjecture on the intersection of weighted badly approximable vectors and use this to prove an instance of a conjecture of An, Guan and Kleinbock. Namely, let $G := SL_2(\mathbb{R}) \times…

This paper improves some results of the author's previous work. We will investigate the case of non-smooth points on an automorphic components and prove Breuil-Schenider conjecture. As a consequence we will see that in case when all the…

The Tur\'{a}n number of a graph $H$, denoted $\mbox{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph with no subgraph isomorphic to $H$. Solymosi conjectured that if $H$ is any graph and $\mbox{ex}(n,H) = O(n^{\alpha})$…

Let BS(m,n) denote the set of base sequences (A;B;C;D), with A and B of length m and C and D of length n. The base sequence conjecture (BSC) asserts that BS(n+1,n) exist (i.e., are non-empty) for all n. This is known to be true for n <= 36…

We prove the automorphism conjecture for ordered sets of width less than or equal to 11. The proof supports the meta conjecture that a large number of automorphisms is achievable only as some type of product of independent automorphisms on…

We show that there exists an absolute constant $c_0<1$ such that for all $n \ge 2$, any measurable set $A \subset S^{n-1}$ of density at least $c_0$ contains $n$ pairwise orthogonal vectors. The result is sharp up to the value of the…

A conjecture of Erd\H{o}s states that for any infinite set $A \subseteq \mathbb R$, there exists $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for…

The absolute sets of local systems on a smooth complex algebraic variety are the subject of a conjecture of N. Budur and B. Wang based on an analogy with special subvarieties of Shimura varieties. An absolute set should be the…

We construct a weakly compact convex subset of $\ell^2$ with nonempty interior that has an isolated maximal element, with respect to the lattice order $\ell _+^2$. Moreover, the maximal point cannot be supported by any strictly positive…

For an $n$-vertex graph $G$, let $h(G)$ denote the smallest size of a subset of $V(G)$ such that it intersects every maximum independent set of $G$. A conjecture posed by Bollob\'{a}s, Erd\H{o}s and Tuza in early 90s remains widely open,…

In this article, we show the existence of large sets $\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of…

Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded…