In 1983, Daykin and Frankl conjectured that if $P$ is a convex subset of $Q_n$, then it contains at least $|P|\binom{n}{\lfloor n/2\rfloor}2^{-n}$ pairwise incomparable elements. We verify and communicate an LLM-generated proof of this conjecture.
We show that there exists a constant $\epsilon>0$ such that if $C$ is a conjugacy class in $A_n$ of size at least $|A_n|^{1-\epsilon}$, then $A_n \setminus \{1\} \subseteq C^2$. This proves the robust version of Thompson's conjecture for the alternating group, conjectured by Shalev (Annals of Math., 2009). Our proof co...
We prove that for every acyclic matrix pattern $P$ it holds that $\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$.
In this short note, we show that the loss of order $N^{c_p/\log\log N}$ in the version of the Ionescu--Wainger multiplier theorem for canonical fractions, recently established by Kosz, Mirek, S{\l}omian, and Zhang in arXiv:2606.12663, is optimal.
We prove that the Esseen's constant $0.4097...$ is valid in the iid Berry--Esseen inequality as long as the sample sizes $n\ge N$. The threshold $N$ is universal over all real summand distributions with positive variance and a finite third absolute moment. In particular, one may take $N=2\left\lceil\exp(10^{17})\right\...
We provide a counterexample to the Pierce-Birkhoff conjecture: a continuous function that is piecewise quadratic on a finite collection of semialgebraic sets that partition $\mathbb{R}^n$ but that cannot be expressed as a finite lattice combination of polynomials. The counterexample was found with the assistance of our...
Fishburn's latent-subset conjecture, proposed in 1987 and revisited in 1988, asserts that for every dual intersecting family $\mathcal F\subseteq 2^{[n]}$, there exists $i\in[n]$ such that $|\mathcal F^L(i)|\geq|\mathcal F(i)|$. Here $\mathcal F^L$ is the family of the subsets of members of $\mathcal F$ that do not bel...
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