We give a general principle that converts fixed-uniformity bounds for forbidden traces into bounds with polynomial dependence on the uniformity. More precisely, let $H$ be a fixed set system on $h\ge1$ vertices, and suppose that, for some $\alpha\ge0$, $\operatorname{ex}_j(m,\operatorname{Tr}(H))=O_{H,j}(m^\alpha)$ for every fixed integer $j\ge1$. Then, for every $\varepsilon>0$, there is a constant $C_{H,\alpha,\varepsilon}$ such that \[ \operatorname{ex}_r(m,\operatorname{Tr}(H)) \le C_{H,\alpha,\varepsilon} r^{h-1-\alpha+\varepsilon}m^\alpha \] for all $m\ge r\ge2$. In particular, for trace-$C_4$-free hypergraphs and every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that \[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \le C_\varepsilon r^{3/2+\varepsilon}m^{3/2} \] for all $m\ge r\ge2$. We also construct trace-$C_4$-free $r$-graphs showing that \[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \ge c r^{1/2}m^{3/2} \] for an absolute constant $c>0$, for every fixed $r\ge3$ and all sufficiently large $m$ (depending on $r$).
For an $r$-uniform hypergraph $H$, let $\nu(H)$ be the maximum number of edges no two of which share $r-1$ vertices, and $\tau(H)$ the minimum number of $(r-1)$-sets such that every edge contains one of them. Aharoni and Zerbib conjectured that $\tau(H)\le\lceil\frac{r+1}{2}\rceil\,\nu(H)$, which for $r=3$ generalizes...
An $r$-uniform hypergraph is linear if any two edges share at most one vertex. For a family $\mathcal F$ of $r$-graphs, let $\text{ex}^{\text{lin}}_r(n,\mathcal F)$ and $\text{spex}^{\text{lin}}_r(n,\mathcal F)$ denote, respectively, the maximum number of edges and the maximum spectral radius of an $n$-vertex $\mathcal...
Let $I$ be a nonzero proper monomial ideal in ${\rm k}[x_1,\ldots,x_r]$, and let $d$ be the maximal degree of a minimal monomial generator. We prove that the index of stability of the associated primes of its ordinary powers satisfies \[ {\rm astab}(I)\le\frac{(4e)^{r-1}}4\,d^{r-1}\qquad(r\ge3), \] while ${\rm astab}(I...
Consider a constraint satisfaction problem $\mathbf{C}$ on finitely many independent random variables with dependency graph $G$. Let $p_a$ be the violation probability of a constraint $a\in \mathbf{C}$ and $N_G^2 (a)$ the set of constraints at distance one or two from $a$ in $G$. Suppose that, there exists $x\in (0,1)^...
Vishesh Jain, Clayton Mizgerd, H. Pham· 0 citations
For graphs $H$ and $F$, let $\text{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Gerbner, Gy\H{o}ri, Methuku, and Vizer proved that $\text{ex}(n,C_{2\ell},C_{2k})=\Theta(n^\ell)$ for $k>\ell\ge2$. They determined the leading term for $\ell=2$, but for $k>\ell\ge3$ their general...
Let $\chi_m$ be a primitive real-valued Dirichlet character of conductor $m\ge3$, let $L(s)=L(s,\chi_m)$ be the associated Dirichlet $L$-function, and let $\xi_L(s)$ denote its completed form. For each $t \in \mathbb R,$ we define the entire function \[ H^L_t(z) := \int_{-\infty}^{\infty} e^{tu^2}\, \Phi_L(u)\, e^{izu}...
R. Freire, Cristian Gonz'alez-Riquelme, Claudio Muñoz et al.· 0 citations
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