Let $g(n)$ denote the largest number of facets of a full-dimensional $0/1$-polytope in $\R^n$. We prove that there are absolute constants $c>0$ and $n_0$ such that $$ g(n)\ge (cn)^{n/2}\quad(n\ge n_0). $$ This removes the logarithmic factor from the lower bound $\bigl(cn/\log n\bigr)^{n/2}$ of Gatzouras, Giannopoulos, and Markoulakis. The proof compares a random sign polytope with two Rademacher rate bodies separated by a fixed level gap. Facets missing the inner body have uniformly small footprints on a flat patch of the outer body. A facet entering the inner body forces an empty buffered discrete cap. For shallow penetration, a likelihood-slab localization reduces the relevant range entropy and permits a conditional $\varepsilon$-net argument; for deep penetration, a global discretization suffices.
A long-standing question posed by Fukuda (1995) and Ziegler (2000) inquires about the asymptotic behavior of $g(n)$, the maximum number of facets that an $n$-dimensional $0/1$-polytope can have. A remarkable result by B\'ar\'any and P\'or (2001) via probabilistic methods established that $g(n)$ is at least superexponen...
We prove bounds of order $n^{n/2}e^{O(n)}$ for the expected number of facets of high-dimensional random polytopes. First, let $\mu$ be a non-degenerate compactly supported even probability measure on $\R$ satisfying $\mu([x^\ast-s,x^\ast])\asymp s^\kappa$ near its right endpoint $x^\ast$. For every sufficiently small f...
Let $D(N)$ denote the largest cardinality of a subset of $\{1,\ldots,N\}$ containing no nonzero square difference. While a construction certifying $D(N)\geq (1-o(1))N^{1/2}$ is almost trivial, Erd\H{o}s conjectured that this bound is sharp up to polylogarithmic factors. This was disproved by S\'ark\"ozy and later again...
For a graph $F$, the Tur\'an number $\operatorname{ex}(n,F)$ is the maximum number of edges in an $n$-vertex graph containing no copy of $F$. Determining the Tur\'an numbers of even cycles is a central problem in extremal graph theory and remains open in general. For $C_6$, the best previous upper bound was due to F\"u...
Sandip Das, Sk Samim Islam, A. Mohapatra et al.· 0 citations
For positive integers $n\ge s>r$, let $T(n,s,r)$ denote the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices such that every $s$-set of vertices contains at least one edge. A simple averaging argument shows that the ratio $T(n,s,r)/\binom nr$ is non-decreasing in $n$ and we denote its limit as $n\to...
Jun Gao, Pei-Ru Kuang, Oleg Pikhurko et al.· 0 citations
A family of subsets of $[n]$ is $r$-fork-free if none of its members is strictly contained in $r$ other distinct members. For each fixed integer $r\ge2$, we prove that the maximum size of such a family is \[ \binom{n}{\lfloor n/2\rfloor} \left(1+\frac{2(r-1)}{n}+o(n^{-1})\right). \] This determines the second-order ter...
Yi-Yang Zhan, Mei Lu, Xia-Miao Zhao· 1 citation
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