arXiv subjects

Yayi Fu

Publications and source records attributed to Yayi Fu.

6 recordsLinked to original sources

arXiv2026

Enclosing a Compact Set in an O-minimal Expansion of $(\mathbb{R},+,\cdot,0,1 <)$

Fix an o-minimal expansion $\mathcal{R}=(\mathbb{R},+,\cdot,0,1 <,...)$ of the real ordered field. Given $C^1$ functions $f_1,...,f_k$, $g_1,...,g_k:M\to\mathbb{R}$ on a definable cell $M$, let $h_{i,0}$ denote $f_i$ and $h_{i,1}$ denote $g_i$. Suppose that for all $\tau\in 2^{[k]}$, $H_{\tau}=(h_{1,\tau(1)},..,h_{k,\tau(k)}) :M\to \mathbb{R}^k$ is regular and proper on $M$, and that for all $i\in[k]$, $\{f_i=0\}$ and $ \{g_i=0\}$ are connected, and $\{f_i=0\}\cap \{g_i=0\}=\emptyset$. We show that then there exists a sequence $(\square_{i,\epsilon}:i\in[k],\epsilon \in \{0,1\})\in\{\leq ,\geq \}^{[k]\times\{0,1\}}$ such that the enclosed region $\underset{i\in[k]}{\bigcap}\{ f_i\square_{i,0} 0\}\cap \{ g_i\square_{i,1} 0\} $ is compact.

math.LO↗

arXiv2026

Towards Trans-Exponential O-minimal Expansion of $(\mathbb{R},+,\cdot, 0, 1 <)$

We add an analytic trans-exponential function $\varphi$ to $\mathbb{R}_{an,\exp}$. We reduce the o-minimality of $\mathbb{R}_{an,\exp,\varphi}$ to the existence of "many" regular values for some definable systems of functions, which is a necessary condition for the o-minimality of $\mathbb{R}_{an,\exp,\varphi}$.

math.LO↗

arXiv2023

A model theoretic proof for o-minimal coherence theorem

Bakker, Brunebarbe, Tsimerman showed in \cite{bakker2022minimal} that the definable structure sheaf $\mathcal{O}_{\mathbb{C}^n}$ of $\mathbb{C}^n$ is a coherent $\mathcal{O}_{\mathbb{C}^n}$-module as a sheaf on the site $\underline{\mathbb{C}^n}$, where the coverings are finite coverings by definable open sets. In general, let $\mathcal{K}$ be an algebraically closed field of characteristic zero. We give another proof of the coherence of $\mathcal{O}_{\mathcal{K}^n}$ as a sheaf of $\mathcal{O}_{\mathcal{K}^n}$-modules on the site $\underline{\mathcal{K}^n}$ using spectral topology on the type space $S_n(\mathcal{K})$. (Here $S_n(\mathcal{K})$ means $S_{2n}(\mathcal{R})$ for some real closed field $\mathcal{R}$.) It also gives an example of how the intuition that sheaves on the type space are the same as sheaves on the site with finite coverings (see \cite[Proposition~3.2]{edmundo2006sheaf}) can be applied.

math.LO↗

arXiv2023

Towards Erd\H{o}s-Hajnal property for dp-minimal graphs

We introduce the notion of strongly $\binom{k}{2}$-free graphs, which contain dp-minimal graphs. We show that under some sparsity assumption, given a rainbow $\binom{k}{2}$-free blockade we can find a rainbow $\binom{k-1}{2}$-free blockade. This might serve as an intermediate step towards Erd\H os-Hajnal property for dp-minimal graphs.

math.CO↗

arXiv2023

A note on strong Erd\H{o}s-Hajnal for graphs with bounded VC-minimal complexity

Inspired by Adler's idea on VC minimal theories \cite{adler2008theories}, we introduce VC-minimal complexity. We show that for any $N\in\mathbb{N}^{>0}$, there is $k_N>0$ such that for any finite bipartite graph $(X,Y;E)$ with VC-minimal complexity $< N$, there exist $X'\subseteq X$, $Y'\subseteq Y$ with $|X'|\geq k_N |X|$, $|Y'|\geq k_N |Y|$ such that $X'\times Y' \subseteq E$ or $X'\times Y'\cap E=\emptyset$.

math.LO↗

arXiv2023

A note on Erd\H{o}s-Hajnal property for graphs with VC dimension $\leq 2$

Using techniques in \cite{chudnovsky2023erdHos} and substitution in \cite{alon2001ramsey}, we show that there is $\epsilon>0$ such that for any graph $G$ with VC-dimension $\leq 2$, $G$ has a clique or an anti-clique of size $\geq |G|^\epsilon$. We also show that Erd\H{o}s-Hajnal property of VC-dimension $1$ graphs can be proved using $\delta$-dimension technique in \cite{chernikov2018note}, and we show that when $E$ is a definable symmetric binary relation, \cite[Theorem 1.3]{chernikov2018note} can be proved without using Shelah's 2-rank..

math.LO↗