Related papers: On the Frankl's union-closed conjecture
In this paper, we prove that all finite solvable groups satisfy the Isaacs-Seitz conjecture namely the derived lenght of a finite solvable group G is less than or equal to the number of distinct irreducible complex character degrees of G.
We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked ``prime by prime". For these groups, we show that a strong…
In this paper we formulate and lay the foundations for the K-theoretic Farrell-Jones Conjecture for the Hecke algebra of totally disconnected groups. The main result of his paper is the proof that it passes to closed subgroups. Moreover, we…
The Ramsey Choice principle for families of $n$-element sets, denoted $\mathrm{RC}_n$, states that every infinite set $X$ has an infinite subset $Y\subseteq X$ with a choice function on $[Y]^n := \{z\subseteq Y : |z| = n\}$. We investigate…
The FPP conjecture, proposed by J. Adams, S. Miller, and D. Vogan and proved by D. Davis and L. Mason-Brown in arXiv:2411.01372, imposes a strong upper bound on the infinitesimal character of a unitary representation of a real reductive…
Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak{c}$ if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if…
Gregory McColm conjectured that positive elementary inductions are bounded in a class K of finite structures if every (FO + LFP) formula is equivalent to a first-order formula in K. Here (FO + LFP) is the extension of first-order logic with…
We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if…
We develop some basic results about full amalgamation classes with intrinsic trascendentals. These classes have generics whose models may have finite subsets whose intrinsic closure is not contained in its algebraic closure. We will show…
The Schinzel Hypothesis is a conjecture about irreducible polynomials in one variable over the integers: under some standard condition, they should assume infinitely many prime values at integers. We consider a relative version: if the…
We investigate the relation of countable closed subsets of the reals with respect to continuous monotone embeddability; we show that there are exactly aleph_1 many equivalence classes with respect to this embeddability relation. This is an…
A classical theorem of Wonenburger, Djokovic, Hoffmann and Paige states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give…
We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a…
We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such as $n=3$ or $4$ this gives a new proof of…
A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the…
Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then $h(x)\in \C[x]$ must also have boundedly many terms. Solving an older conjecture raised by R\'enyi and by…
We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erd\H{o}s-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than…
A collection of sets $\Fscr$ has the $(p,q)$-property if out of every $p$ elements of $\Fscr$ there are $q$ that have a point in common. A transversal of a collection of sets $\Fscr$ is a set $A$ that intersects every member of $\Fscr$.…
The main theorem of this article is that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is…
New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…