Related papers: Proof of Union-Closed Sets Conjecture
A very simple but useful almost sure convergence theorem of probability is given.
We provide some examples which give evidence to the conjectures contained in my paper "Finiteness of $p$-Divisible Sets of Multiple Harmonic Sums" (math.NT/0303043). All the main theoretical results can be found in that paper.
The union-closed sets conjecture states that in any nonempty union-closed family $\mathcal{F}$ of subsets of a finite set, there exists an element contained in at least a proportion $1/2$ of the sets of $\mathcal{F}$. Using the…
In this paper, we prove certain theorems about three consecutive primes.
In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.
We demonstrate the truth of the sunflower conjecture by showing that a family $\mathcal{F}$ of sets each of cardinality at most $m$ includes a $k$-sunflower, if $|\mathcal{F}| > ( c k )^{2m}$ for a constant $c>0$ independent of $m$ and $k$,…
The paper presents a counterexample to the Hodge conjecture.
The article provides a counterexample to a conjecture by Blocki-Zwonek.
We prove the Burghelea Conjecture for groups satisfying some additional cohomological property.
We describe recent advances in the study of random analogues of combinatorial theorems.
The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\frac{1}{2}\cdot m$ sets in $\mathcal{F}$. In 2022,…
We prove tight closure analogues of results of Watanabe about chains and families of integrally closed ideals.
We prove bounds for the covering numbers of classes of convex functions and convex sets in Euclidean space. Previous results require the underlying convex functions or sets to be uniformly bounded. We relax this assumption and replace it…
It is proved that the continuum hypothesis implies the existence of a group M containing a nonalgebraic unconditionally closed set, i.e., a set which is closed in any Hausdorff group topology on M but is not an intersection of finite unions…
We prove the Farrell-Jones conjecture for free-by-cyclic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture.
We prove a discrete approximation of functionals with jumps and creases.
We prove the topological analogue of the period-index conjecture in each dimension away from a small set of primes.
In this paper, we prove and disprove several generalizations of unbounded versions of the Fuglede-Putnam theorem.
We establish coupled fixed point theorems for contraction involving rational expressions in partially ordered metric spaces.
Mathematicians had little idea whether the easy-to-state union-closed conjecture was true or false even after $40$ years. However, last winter saw a surge of interest in the conjecture and its variants, initiated by the contribution of a…