Related papers: Selection principles and proofs from the Book
We extend Berge's Maximum Theorem to allow for incomplete preferences. We first provide a simple version of the Maximum Theorem for convex feasible sets and a fixed preference. Then, we show that if, in addition to the traditional…
The basic results of a new theory of regular functions of a quaternionic variable have been recently stated, following an idea of Cullen. In this paper we prove the minimum modulus principle and the open mapping theorem for regular…
For an upper semi-continuous set-valued mapping from one topological space to another and for a lower semi-continuous function defined on the product of these spaces, Berge's theorem states lower semi-continuity of the minimum of this…
The topological Tverberg theorem claims that for any continuous map of the (q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images have a non-empty intersection. This has been proved for affine maps, and if $q$ is a…
This paper concerns two-player alternating play combinatorial games (Conway 1976) in the normal-play convention, i.e. last move wins. Specifically, we study impartial vector subtraction games on tuples of nonnegative integers (Golomb 1966),…
A longstanding conjecture of Seymour states that in every oriented graph there is a vertex whose second outneighbourhood is at least as large as its outneighbourhood. In this short note we show that, for any fixed $p\in[0,1/2)$, a.a.s.…
In this note, we compare and contrast various selective divergence properties such as the properties of being discretely selective and selectively highly divergent. We identify and incorporate a class of subsemigroups of the semigroup of…
We give new combinatorial proofs of known almost-periodicity results for sumsets of sets with small doubling in the spirit of Croot and Sisask, whose almost-periodicity lemma has had far-reaching implications in additive combinatorics. We…
We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not…
We demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve…
We add to the theory of preservation of topological properties under forcing. In particular, we answer a question of Gilton and Holshouser in a strong sense, showing that if player II has a winning strategy in the strong countable fan…
Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all…
We characterize the computational content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called {\em exactly large} sets. An {\it exactly large} set is a set $X\subset\Nat$ such that $\card(X)=\min(X)+1$.…
In this work we employ machine learning to understand structured mathematical data involving finite groups and derive a theorem about necessary properties of generators of finite simple groups. We create a database of all 2-generated…
In this note we generalize the Moroianu's fixed point theorem. We propose a very elegant common proof of the Bourbaki's fixed point theorem and our result. We apply our result to give a very elegant proof of the fact that, in the…
An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical'…
We prove a selection theorem for paraconvex-valued mappings defined on {\tau}-PF normal spaces. The method developed to prove this result is used to provide a general approach to such selection theorems.
The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemer\'edi theorem which says that any subset of a pseudorandom set of…
The purpose of this note is to prove the existence of a randomized mechanism, a social decision scheme (SDS), with desirable fairness, efficiency, and strategyproofness properties unmatched by all known SDSs. In particular, we disprove a…
We prove the closedness theorem over Henselian valued fields, which was established over rank one valued fields in one of our recent papers. In the proof, as before, we use the local behaviour of definable functions of one variable and the…