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 term and matches the upper bound of De Bonis and Katona. Our lower bound comes from a construction on two adjacent middle levels, using a finite ordered collection of disjoint coordinate blocks and a condition on sums modulo $n$. For each prescribed error, the blocks are fixed before $n$ tends to infinity, so the construction works for every sufficiently large $n$.
For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\le(2\log2+o(1))\,n/\log n$. This improves the bounds of order $n/(\log n)^{1/2-\varepsilon}$ established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We...
A family of permutations in $S_n$ is called even-intersecting if every two distinct members agree in an even number of positions. Let $M(n)$ denote the maximum size of such a family. For even $n$, we prove that $$n!!\leq M(n)\leq e^{\frac{n}{2}+o(n)}n!!,$$ improving the bound obtained from a theorem of Cameron, Deza an...
Anirban Banerjee, Abisek Dewan, R. Mishra· 0 citations
Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$. Erd\H{o}s, Graham, Ivi\'c and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n)>(1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} We improve upon this by a facto...
For integers $d\geq 3$, let $F_{2,d}(n)$ be the largest size of a subset of $[n]$ containing no two distinct elements whose product is a perfect $d$-th power, and let $f_{2,d}(n)$ denote the analogous quantity when the two elements need not be distinct. Fleiner, Juh\'asz, K\"ov\'er, Pach, and S\'andor proved that both...
Fix positive integers $N,k,n$ with $n\ge k$. We seek the minimum number of members of size at least $n$ in a finite family of finite sets closed under union and containing exactly $N$ distinct sets of size $k$. This problem is a specialization of the Leck--Roberts--Simpson weighted conjecture: assign weight one to sets...
For a positive integer $n$, let $S(n)=\{\sigma(d): d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding...
Felix Huber· 0 citations
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