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so we just have to prove $\sum_{S\in\mathcal{F}}\ln w(S)\geq \dots\frac{|\mathcal{F}|\ln|\mathcal{F}|}{2}$ This inequality does not depend on the choice of $a_i$'s, and, unfortunately, it can fail. …

Let $n=|\mathcal F'|$. Below we easily show that $n\ge\sqrt{m}$ and I shall look for more refined arguments to improve this bound. For each natural $N$ put $[N]=\{1,2,\dots,N\}$. For each $i\in [2]$ p …

I am still studying relevant sources, trying to provide the required bounds. So I am going to update this answer accordingly. If $l>k$ or $r=1$ then the required condition always holds, so we assume t …

Fedor Petrov's comment shows than for each $k$ there are only finitely many cases for $x_1$ to check, so I wrote a program to do this. The positive answer is already obtained for all natural $k\le 10$ …

Let us show that a required $k$-times bin-packing exists for some $k\le n^{n/2}$. Let $V=\{(v_1,\dots,v_n)\in\{0,1\}^n:\sum_{i=1}^n d_iv_i\le s\}$ be the set of all admissible "bin packings". Then the …

I tried to find a required example, but failed (I share my findings below). Nevertheless, it seems rather strange to me if there is no such example, so I hope that it can be constructed. A natural ide …

Proposition. $f(n)\ge 2n+\tfrac 13\left(\sqrt{\tfrac{n}3+1}-2\right) $ for any natural $n\ge 13$. Proof. Fix the cake cutting with the minimum number $f=f(n)$ of slices. We shall work with the graph $ …

This question is related to my last question and is originally motivated by recent advances in quantum physics. I am looking for combinatorial characterizations of some algebraically defined families …

In 2018 Mario Krenn posed this originated from recent advances in quantum physics question on a maximum number of colors of a monochromatic graph with $n$ vertices. Despite very intensive Krenn’s prom …

Your suspicion is correct. The following hypergraph $H$ provides a negative answer to your question. Let $V=\{0,1,\dots, 11\}$. Then $V=V_0\cup V_1\cup V_2$, where $V_0=\{0,1,2,3\}$, $V_1=\{4,5,6,7\}$ …

According to [p. 8], Baranyai's theorem [B] implies that the vertex set of the Kneser graph $K(n,k)$ can be partitioned into $\left\lceil\frac{\binom{n}{k}}{\left\lfloor\frac{n}{k}\right\rfloor}\righ …

This answer is partial. We assume that all $a_i$ are non-negative. Put $A=\tfrac 1m \sum a_i$ and $A_j=\sum_{i\in S_j} a_i$. We want to maximize $\Pi=\prod A_j$. But by AM-GM inequality, $P^{1/m}\le …

For every family $\mathcal F\subset\mathcal P_f(\Bbb N) $ both a set $\{1, 3,4, 7,8,9, 13,14,15,16,\dots\} $ and its complement have upper Banach density $d^*_{\mathcal F}$ equal to $1$.

For each $1\le i\le n-1$ the sum of numbers of $i$-th column of the lower triangle $\Delta$ below the main diagonal is at least $1+\dots+i=i(i+1)/2$. Thus the sum of all numbers in $\Delta$ is at leas …

There is a special version of this question (for $n=15$ and $m\le 7$) at Mathematics.SE. For each $n\ge 2$ I constructed a set $S$ with the property requiring $m\ge \left\lfloor\tfrac n2\right\rfloor= …