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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 $ …

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. …

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 …

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 …

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 $V_0=\{0,1,2,3\}$, $V_1=\{4,5,6,7\}$ …

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 …

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 …

Wagner-Fáry-Stein theorem states that each (finite simple) planar graph admits a straight-line crossing-free plane drawing. On the other hand, each graph (of at most $\frak c$ vertices) admits a strai …

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 $1$.

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 …

Naturally generalizing your example we can show that $\alpha=0$. Consider the following construction. For any natural number $k$ put $n=2^k$ and consider matrices $A_1,\dots, A_k$, where each $A_j$ c …

It seems the following. I can propose the following example with a not so trivial lower bound. Exactly, for $l\ge 2$ I have $m=2^{l-1}$, $n=2^{l-1}(2^l-1)$ and $N=2^l-1$. Let $G=\Bbb Z_2^l$ be an $l …

I asked this question at MSE some months ago but I received only partial answers, so I put it here. The following sounds nice for me and I spent a good time during the investigation. But I am a profes …

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= …

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 $k\le 10$ …