arXiv subjects

Dean Menezes

Publications and source records attributed to Dean Menezes.

5 recordsLinked to original sources

arXiv2026

The Three-Dimensional Erdős Box Problem Has Exponent $11/4$

The Zarankiewicz problem for $3$-uniform hypergraphs asks for the maximum number $z(n)$ of edges in a tripartite hypergraph with $n$ vertices in each part containing no copy of $K_{2,2,2}^{(3)}$ (a ``box''). Erdős (1964) proved $z(n) = O(n^{11/4})$. The best previously known lower bound was $Ω(n^{8/3})$, due to Katz, Krop, and Maggioni (2002). We construct a family of box-free hypergraphs matching Erdős's upper bound: for each $q = 2^m$ ($m \geq 1$), our hypergraph has $q^4$ vertices in each part and $q^{11}$ edges, establishing that $z(n) = Θ(n^{11/4})$. The construction is algebraic, defined over ${\bf F}_{q^3}$ via the power map $τ(s) = s^{q^2-q+1}$. The proof shows that the direction-$1$ finite differences of $τ$ partition ${\bf F}_{q^3}$ into pairwise skew affine lines over ${\bf F}_q$, preventing boxes from forming.

math.CO↗

arXiv2026

A Lorentz Endpoint and Quantitative Obstruction for a Cowling-Price Moment Reduction

Let $$ M_{r,p}^{(d)}(g)=\int_{\mathbb R^d}|x|^r|g(x)|^p\,dx, \qquad 1 d(p-1)$. We then quantify the transition: the best constant blows up like $[r-d(p-1)]^{-1/p'}$ at the critical line, while on the annulus $1\le |x|\le R$ the endpoint loss is exactly $(dω_d\log R)^{1/p'}$. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls $L^{1,p}$, $$ \lVert g\rVert_{L^{1,p}}\le ω_d^{1/p'}M_{d(p-1),p}^{(d)}(g)^{1/p}, $$ and strengthening the weighted norm from $L^p$ to $L^{p,1}$ restores the strong target, $$ \lVert g\rVert_1\le ω_d^{1/p'} \lVert |x|^{d/p'}g\rVert_{L^{p,1}}. $$ More generally, every power moment has a sharp target Lorentz space, and its secondary index is optimal. These results identify the endpoint lost by the direct same-$q$ reduction. They do not obstruct auxiliary-norm versions of the Wigderson framework, which reach the Cowling--Price range by a different route.

math.GM↗

arXiv2026

Sharp Bounds for Discrete Cube Skeleta

Fix integers $0\leq k<n$. Let $F_{n,k}(N)$ be the least size of a finite set $B\subset\Z^n$ that contains a filled axis-parallel cube $k$-skeleton centered at each point of some $N$-point set. We prove that $F_{n,k}(N)$ has order $N^{1-(n-k)/(2n^2)}$, with constants depending only on $n$ and $k$. Thornton proved the upper bound and lower bounds with every smaller exponent; the endpoint lower bound was open for $k\geq1$. For square boundaries in $\Z^2$, the exponent is $7/8$. A midpoint count and Shearer's inequality handle large radii; induction in lattice cells handles small radii.

math.CO↗

arXiv2026

Characteristic drops for high-order vanishing on the hypercube

Let $F$ be a field, let $0\le \ell\le k-2$, and suppose that $n\ge k-1$. We determine the minimum degree of a polynomial in $F[x_1,\ldots,x_n]$ that vanishes to order at least $k$ at every nonzero vertex of the Boolean cube and to order exactly $\ell$ at the origin. The answer is \[ n+2k-2-ρ_F(k-\ell), \] where $ρ_F(s)$ is the least number of positive integers summing to $s-1$ whose corresponding Catalan numbers are nonzero in $F$. The proof gives an explicit basis of the reduced vanishing space. In this basis the top-degree map is diagonal, with Catalan numbers on the diagonal; a refinement using block coordinates shows that the basis is compatible with polynomial degree. In odd characteristic the degree is either $n+2k-3$ or $n+2k-4$, according as $C_{k-\ell-2}$ is nonzero or zero. In characteristic $2$ it is \[ n+2k-2-s_2(k-\ell-1), \] where $s_2$ denotes binary digit sum. Thus the first characteristic drop, and all later drops, are determined exactly.

math.CO↗

arXiv2026

A Direct Proof of the Locally Dense Graphon Inequality

Bradač, Sudakov, and Wigderson characterized $p$-locally dense graphons by a quadratic inequality for all bounded nonnegative functions. Their proof uses Reiher's finite lemma and graphon approximation, and they asked for a direct proof. We give one by rounding simple functions. Divide each level set into $m$ equal-measure pieces and retain each piece independently with probability equal to its level. Off-diagonal terms agree in expectation; the diagonal error is at most $\norminf{W-p}/(4m)$. Letting $m\to\infty$ and then approximating in $L^1$ proves the inequality. A three-atom example shows that atomlessness is necessary if arbitrary probability spaces are allowed.

math.CO↗