Related papers: Proof of Union-Closed Sets Conjecture

We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$.

We prove some 3-adic congruences for binomial sums, which were conjectured by Sun.

In this paper, we prove a conjecture proposed by George Beck, which involves gap-free partitions and partitions with distinct parts.

Most of the assertions in the theory of well ordered sets are quite simple. However, one of its central statements, Zermelo's theorem, stands out of this rule, for its well-known proofs are rather complicated. The aim of the current paper…

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups.

Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.

We proved that there are infinitely many cousin primes.

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can…

We improve the best known constant $\frac{3-\sqrt 5}{2}$ for which the union-closed conjecture is known to be true, by using dependent samples as suggested by Sawin and the entropy approach on this problem initiated by Gilmer. Meanwhile, we…

In this paper, we give a proof of the Bouchard-Klemm-Marino-Pasquetti conjecture for a framed vertex, by using the symmetrized Cut-Join Equation developed in a previous paper.

We prove that the following problem is decidable: given a finite set of relations, decide whether this set admits a near-unanimity function.

The Frankl's conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. FC-families are families for which it is proved that every…

For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a…

We prove Manin's conjecture for bi-equivariant compactifications of unipotent groups.

New cases of the multiplicity conjecture are considered.

An technically interesting proof of a known theorem.

We present a conjecture about partitions, with a very elementary formulation.

We prove a variation of Gronwall's lemma.

Talagrand conjectured that if a family of sets $\mathcal{F}$ over $X = \{ 1,2,\cdots, N \}$ is of large measure, then constant times of unions of sets in $\mathcal{F}$ will cover a large portion of the power set of $X$. This conjecture is a…

We prove that the non-vanishing conjecture holds for generalized lc pairs with a polarization.