Related papers: Proof of Union-Closed Sets Conjecture
In this short note, we prove Hadwiger's conjecture for strongly monotypic polytopes.
We present a proof of a combinatorial conjecture from the second author's Ph.D. thesis. The proof relies on binomial and multinomial sums identities. We also discuss the relevance of the conjecture in the context of PAC-Bayesian machine…
In this note, we provide a short proof of Feige's conjecture for identically distributed random variables.
Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.
We prove the positivity conjecture for all skew-symmetric cluster algebras.
Under the prime-tuple hypothesis, the set of signed primes is a sumset.
Based on the results people have obtained, we try to prove the Jacobian conjecture, but there is a gap in the proof.
The Union-Closed Sets Conjecture asks whether every union-closed set family $\mathcal{F}$ has an element contained in half of its sets. In 2022, Nagel posed a generalisation of this problem, suggesting that the $k$th-most popular element in…
The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can…
Here we prove some conjectures on the monotony of combinatorial sequences from the recent preprint of Zhi--Wei Sun.
A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions.
We prove a connectedness result for products of weighted projective spaces.
We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.
Given a union-closed family $\mathcal{F}$ of subsets of the universe $[n]$, with $\mathcal{F}$ not equal to the power set of $[n]$, a new subset $A$ can be added to it such that the resulting family remains union-closed. We construct a new…
A vector variational principle is proved.
We give a short and uniform proof of a special case of Tits' Centre Conjecture using a theorem of J-P. Serre and a result from our earlier work. We consider fixed point subcomplexes $X^H$ of the building $X = X(G)$ of a connected reductive…
We give a proof of some small weight and level cases of Serre's conjecture.
In this paper we give a complete proof of the Brumer-Stark conjecture over $\mathbf{Z}$.
In this paper, we give a simple counter example to the famous Hodge conjecture.
We give a new proof of a classical theorem on approximation of continuous functions on totally real sets