Related papers: Proof of Union-Closed Sets Conjecture
We illustrate the concept of mathematical proof.
We prove a version of $Z$-set unknotting theorem for uncountable products of real numbers.
We use the recently introduced \'etale open topology to prove several facts about large fields. We show that these facts lift to a very general topological setting.
We prove several congruences for trinomial coefficients.
Proofs of Tychonoff's theorem often seem to require a bit of magic. Machinery such as ultrafilters, nets or maximal families with the finite intersection property are employed to give proofs that can be very neat, but not the kind of thing…
We introduce a recursive method to deconstruct the automorphism group of an ordered set. By connecting this method with deep results for permutation groups, we prove the Automorphism Conjecture for ordered sets of width less than or equal…
We identify a number of decidable and undecidable fragments of first-order concatenation theory. We also give a purely universal axiomatization which is complete for the fragments we identify. Furthermore, we prove some normal-form results.
We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations.
Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.
We prove a partition identity conjectured by Lassalle (Adv. in Appl. Math. 21 (1998), 457-472).
We prove the "strong conjecture" expressed by Gazeau et al. in arXiv:1203.3936v1 [math-ph] about the coefficients of the Taylor expansion of the exponential of a polynomial. This implies the "weak conjecture" as a special case. The proof…
We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of…
We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways of computing the correlation functions of a discrete orthogonal polynomial…
In this work we resolve several conjectures stated in the On-Line Encyclopedia of Integer sequences.
We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called supratopological spaces. We prove some results on the…
The purpose of this article is to present my new proof of the the construction and the convergence theorem of spectral sequences of filtered complexes, which is much shorter and cleaner than the "standard" proof.
We prove a local version of the Mazur-Ulam theorem.
Suppose that $\mathscr{F}$ is a finite union-closed family of sets with $\cup_{A\in \mathscr{F}}A=\{1,2,\ldots,m\}$ and $m\geq 2$. Fix $i\in \{1,2,\ldots,m\}$ and denote $\mathscr{G}:=\{A\backslash \{i\}: A\in \mathscr{F}\}$. For $j\in…
We prove a sharp structural result concerning finite colorings of pairs in well-founded trees.
We prove an infinite family of lacunary recurrences for the Lucas numbers using combinatorial means.