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Moody Urbanity - Relations V 🪢
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The maximum product of sizes of cross-\(t\)-intersecting families
Jingjun Bao, Lijun Ji
https://arxiv.org/abs/2510.11724 https://arxiv.org/pdf/2510.11724…

The maximum product of sizes of cross-\(t\)-intersecting families
Two families of sets \(\mathcal{A}\) and \(\mathcal{B}\) are called \emph{cross-\(t\)-intersecting} if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\). Determining the maximum product of sizes for such cross-\(t\)-intersecting families is an active problem in extremal set theory. In this paper, we verify the following cross-\(t\)-intersecting version of the Erdős-Ko-Rado theorem: For \(k\geq l \geq t \geq 3\) and \(\min\{m,n\} \geq (t+1)(k-t+1)\), the ma…

On irreducibility of certain low dimensional automorphic Galois representations
Boyi Dai
https://arxiv.org/abs/2510.12496 https://arxiv.org/pdf/2510.12496

On irreducibility of certain low dimensional automorphic Galois representations
We study irreducibility of Galois representations $ρ_{π,λ}$ associated to a $n=7$ or 8-dimensional regular algebraic essentially self-dual cuspidal automorphic representation $π$ of $\text{GL}_n(\mathbb{A}_\mathbb{Q})$. We show $ρ_{π,λ}$ is irreducible for all but finitely many $λ$ under the following extra conditions. (i) If $n=7$, and there exists no $λ$ such that the Lie type of $ρ_{π,λ}$ is the standard representation of exceptional group $\textbf{G}_2$. (ii) If $n=8$, and when …

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Urban Adventure 🏞️
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Expansion of trivariate polynomials using proximity
Orit E. Raz
https://arxiv.org/abs/2510.12191 https://arxiv.org/pdf/2510.12191

Expansion of trivariate polynomials using proximity
We extend the proximity technique of Solymosi and Zahl [J. Combin. Theory, Ser. A (2024)] to the setting of trivariate polynomials. In particular, we prove the following result: Let $f(x,y,z)=(x-y)^2+(φ(x)-z)^2$, where $φ(x)\in \mathbb{R}[x]$ has degree at least 3. Then, for every finite $A,B,C\subset \mathbb{R}$ each of size $n$, one has $|f(A,B,C)|=Ω(n^{5/3-\varepsilon})$, for every $\varepsilon>0$, where the constant of proportionality depends on $\varepsilon$ and on ${\rm deg}(…

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