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pro vyhledávání: '"05c55"'
Autor:
Ghosh, Arpita, Ghosh, Surojit
The set of sums of two squares plays an important role in the elementary number theory. Di Nasso investigated several infinite monochromatic patterns in integers considering operations induced from affine maps and asked whether one can find different
Externí odkaz:
http://arxiv.org/abs/2411.14066
Autor:
Lehavi, Adam M.
The Ramsey number $R(s,t)$ is the smallest integer $n$ such that all graphs of size $n$ contain a clique of size $s$ or an independent set of size $t$. $\mathcal{R}(s,t,n)$ is the set of all counterexample graphs without this property for a given $n$
Externí odkaz:
http://arxiv.org/abs/2411.04267
Autor:
Parczyk, Olaf, Spiegel, Christoph
Graham, R\"odl, and Ruci\'nski originally posed the problem of determining the minimum number of monochromatic Schur triples that must appear in any 2-coloring of the first $n$ integers. This question was subsequently resolved independently by Datsko
Externí odkaz:
http://arxiv.org/abs/2410.22024
Let $a_1,\ldots,a_m$ be nonzero integers, $c \in \mathbb Z$ and $r \ge 2$. The Rado number for the equation \[ \sum_{i=1}^m a_ix_i = c \] in $r$ colours is the least positive integer $N$ such that any $r$-colouring of the integers in the interval $[1
Externí odkaz:
http://arxiv.org/abs/2410.16051
A randomly perturbed graph $G^p = G_\alpha \cup G(n,p)$ is obtained by taking a deterministic $n$-vertex graph $G_\alpha = (V, E)$ with minimum degree $\delta(G)\geq \alpha n$ and adding the edges of the binomial random graph $G(n,p)$ defined on the
Externí odkaz:
http://arxiv.org/abs/2410.11003
Autor:
Dobák, Dániel, Mulrenin, Eion
The Erd\H{o}s-Rado canonization theorem generalizes Ramsey's theorem to edge-colorings with an unbounded number of colors, in the sense that for $n = ER(m)$ sufficiently large, any edge-coloring of $E(K_n) \to \mathbb{N}$ will yield some copy of $K_m
Externí odkaz:
http://arxiv.org/abs/2410.08982
Autor:
Marcone, Alberto, Osso, Gian Marco
This paper classifies different fragments of the Galvin-Prikry theorem, an infinite dimensional generalization of Ramsey's theorem, in terms of their uniform computational content (Weihrauch degree). It can be seen as a continuation of arXiv:2003.042
Externí odkaz:
http://arxiv.org/abs/2410.06928
In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs $\mathcal{H}$ in $K_n$, defined as the minimum number of colours needed to colour the edges of $K_n$ so that every copy of a graph $H\in \mathcal{
Externí odkaz:
http://arxiv.org/abs/2410.05887
Autor:
Ding, Guoli, Qualls, Brittian
In this paper we prove that every sufficiently large 4-edge-connected graph contains the double cycle, $C_{2,r}$, as an immersion. In proving this, we develop a new tool we call a ring-decomposition. We also prove that linear edge-connectivity implie
Externí odkaz:
http://arxiv.org/abs/2410.04538
Autor:
Zhang, Yanbo, Chen, Yaojun
As a significant variation of Ramsey numbers, the Gallai-Ramsey number $GR_k(H)$ refers to the smallest positive integer $r$ such that, by coloring the edges of $K_r$ with at most $k$ colors, there exists either a monochromatic subgraph isomorphic to
Externí odkaz:
http://arxiv.org/abs/2410.01549