We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.
We show that there exists $c>0$ such that any subset of $\{1,\ldots,N\}$ having size $\gg N / \exp( (\log\log\log N)^c )$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y^3$. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quan...
Bartłomiej Bychawski, Bartosz Głowacki, Ivan Spyrydonov· 0 citations
Let $G$ be a finite group and $x \in G$. In 2013, Hai-Reuven proved that the cardinality of the centralizer of $x$ divides the cardinality of the solubilizer of $x$. Settling a conjecture of Mousavi, Poozesh and Zamani, we show that the cardinality of the normalizer of $\langle x \rangle$ is also a divisor. We actually...
We give a short proof that $\kappa$-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable $\infty$-categories. A consequence is that an $\infty$-category $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^\mathsf{op}$ are presentable is a...
Thorger Geiß, Phil Pützstück, Maxime Ramzi· 0 citations
The B\'ar\'any-Larman conjecture states that for any $d+1$ sets of $r$ points each in $\mathbb{R}^d$, considered as color classes, we can partition their union into $r$ rainbow $(d+1)$-tuples whose convex hulls intersect. We prove that the topological version of this conjecture holds when $r$ is a prime number. We also...
In this paper, we prove that for any fixed base $b \ge 2$, there exists an infinite arithmetic progression of positive integers $k$ that are simultaneously generalized Sierpi\'nski and generalized Riesel numbers in base $b$. More precisely, we construct such an arithmetic progression so that both $k\cdot b^n + 1$ and $...
We prove that for additive complements $A$ and $B$, if $$ \limsup_{x\to+\infty}\frac{A(x)B(x)}{x}<\frac43, $$ then $$ A(x)B(x)-x\rightarrow+\infty\qquad(x\to+\infty). $$ This improves the earlier upper bound $3-\sqrt{3}$ due to Fang and Chen.
Quan-Hui Yang, Li-Lu Zhao· 0 citations
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