Related papers: Coloring ordinals by reals
We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such…
This paper presents a path to proving the Four-Color Theorem that differs from the traditional "reducible configuration" method. By introducing concepts such as "outer boundary," "primitive set," "Property A," "knot," "valid pair group,"…
Let $k,a\in \mathbb{N}$ and let $p_1,\cdots,p_k\in \mathbb{Q}[n]$ with zero constant term. We show that for any finite coloring of $\mathbb{Q}$, there are non-zero $x,y\in \mathbb{Q}$ such that there exists a color which contains a set of…
We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial…
R\"odl and Ruci\'nski (1990) established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $\hat p_{K_\ell,r}(n)=n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every…
Four-Color Theorem has secret in its logical proof and actual operating. In this paper we will give a proof of Four-Color Theorem based on Kuratowski's Theorem using some induction argument and give a description of the most complicated…
In this paper, we show that Higher-Order Coloured Unification - a form of unification developed for automated theorem proving - provides a general theory for modeling the interface between the interpretation process and other sources of…
In this article we present an axiomatic definition of sets with individuals and a definition of natural numbers and ordinals. We use the axioms pairs, union, power, regularity and separation. We define the equality of sets and of…
The aim of this paper is to give identities which are generalizations of the formulas given by Koornwinder [J. Math. Phys. 30, (1989)] and Hamdi-Zeng [J. Math. Phys. 51, (2010)]. Our proofs are much simpler than and different from the…
We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.
We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation…
We show that under the proper forcing axiom the class of all Aronszajn lines behave like $\sigma$-scattered orders under the embeddability relation. In particular, we are able to show that the class of better quasi order labeled fragmented…
In 1976, Appel and Haken achieved a major break through by proving the four color theorem $(4CT)$. Their proof is based on studying a large number of cases for which a computer-assisted search for hours is required. In 1997, Robertson,…
This paper provides some counterexamples to Cantor's contributions to the foundations of Set Theory. The first counterexample forces Cantor's Diagonal Method (DM) to yield one of the numbers in the target list. To study this anomaly, and…
We prove a generalization of the well-known Bondy-Bopanna-Siegel Lemma to DP-colorings from a class of correspondence assignments which we call derangement assignments. Since DP-colorings from derangement assignments generalize zero-free…
We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.
We generalize a result of Tibor Gallai as follows: for any finite set of points $\mathcal{S}$ in the plane, if the plane is colored in finitely many colors, then there exist $2^{\aleph_0}$ monochromatic subsets of the plane homothetic to…
It is consistent for every (1 <= n< omega) that (2^omega = omega_n) and there is a function (F:[omega_n]^{< omega}-> omega) such that every finite set can be written at most (2^n-1) ways as the union of two distinct monocolored sets. If GCH…
Hypergraph $2$-colorability is one of the classical NP-hard problems. Person and Schacht [SODA'09] designed a deterministic algorithm whose expected running time is polynomial over a uniformly chosen $2$-colorable $3$-uniform hypergraph.…
We review and extend recent findings of Godsil and Zaks, who published a constructive coloring of the rational unit sphere with the property that for any orthogonal tripod formed by rays extending from the origin of the points of the…