Author

D. Menezes

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Preprint Sep 2026

The Three-Dimensional Erd\H{o}s 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''). Erd\H{o}s (1964) proved $z(n) = O(n^{11/4})$. The best previously known lower bound was $\Omega(n^{8/3})$, due t...

D. Menezes · 0 citations

Preprint Aug 2026

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 \[ n+2k-2-\rho_F(k-\ell), \] where $\rh...

D. Menezes · 0 citations