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[Crossposted at math.stackexchange and AoPS]. I would like to prove cases $n=7,8$ of this conjecture (general question asked here): given any commutative semigroup $S$ of order $n \ge 1$, there exist $...
Fabius Wiesner's user avatar
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1 answer
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This "game" is a strengthening of the union-closed sets conjecture (background here and here). Given a positive integer $n \ge 2$ and $m = \lfloor (n-1)/2 \rfloor$, the game starts disposing ...
Fabius Wiesner's user avatar
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10 votes
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Let $\mathcal{F}$ be an union-closed family of subsets of $[n]=\{1,2,...,n\}$, assume $\varnothing\in\mathcal{F}$. Let $l_i=|\{S|S\in\mathcal{F},i\notin S\}|,u_i=|\{S|S\in\mathcal{F},i\in S\}|$, then $...
Veronica Phan's user avatar
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[Crossposted at math.stackexchange] Let $\mathcal{F} = \{\{x_1, x_2\} : 1 \le x_1 \lt x_2 \le n \}$, $n \ge 2$, and let $\mathcal{G} = \{G_1, \ldots, G_n\}$ be a partition of $\mathcal{F}$ in $n$ ...
Fabius Wiesner's user avatar
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I have a problem related to this question and this answer. Consider a graph with $n \ge 3$ vertices, such that any of its induced subgraphs with $m=\lfloor (n+1)/2 \rfloor$ vertices must have at least ...
Fabius Wiesner's user avatar
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3 votes
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Let $S=\{s_1,\ldots,s_n\}$ be any commutative semigroup of order $n \ge 3$. Let $m = \lfloor (n-1)/2 \rfloor$. Consider the system (conjunction) of the following $\binom{n}{2}$ propositions for $1 \le ...
Fabius Wiesner's user avatar
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9 votes
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I would like to solve small cases of this conjecture (general question asked here): given any commutative semigroup $S=\{s_1, \ldots, s_n\}$ of order $n \ge 1$, there exist $a, b \in S$ with $a \ne b$,...
Fabius Wiesner's user avatar
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4 votes
2 answers
240 views

Consider a union-closed family $\mathcal{F} = \{A_1, \ldots, A_n\}$. Define a function: $$e(\mathcal{F}) = \sum_{k=1}^n \binom{|\{\{B,C\}: B,C \in \mathcal{F}, B \cup C = A_k\}|}{2}$$ For example, for ...
Fabius Wiesner's user avatar
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2 votes
2 answers
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Consider a graph $G=(V,E)$ with $V=\{\{a,b\}: a,b \in [n], a \not= b\}$, or equivalently $\ V:=\binom {[n]}2\ $ for short. Assume also that $G$ is a cluster graph (a disjoint union of complete graphs) ...
Fabius Wiesner's user avatar
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Let $G$ be a graph with $\binom{n}{2}$ vertices, each labeled with a different unordered couple $\{a,b\}$ such that $a,b \in [n] = \{1,2,\ldots,n\}, a \not= b$. For any vertex with label $\{a,b\}$ and ...
Fabius Wiesner's user avatar
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I am reading the paper Ramsey-Goodness — and Otherwise by Allen, Brightwell, and Skokan (here is the arXiv version) and I got stuck in the first lemma of section 3 (lemma 19 in the arXiv version and ...
Thaevin's user avatar
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I was reading Even-Zohar’s paper "On sums of generating sets in $\mathbb{Z}_2^n$", which I found very interesting. Here is the link to the arXiv version of the paper. I’m particularly ...
RFZ's user avatar
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I have recently been studying the paper Rainbow Cycles in Properly Edge-Colored Graphs (arXiv:2211.03291), which gives sufficient conditions for the existence of rainbow cycles in properly edge-...
West Book's user avatar
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Let $k$ and $n$ be natural numbers where $0<k\le n$. Let $\mathcal{F}$ be a family of $k$-element subsets of an $n$-element set $[n]:=\{1,2,\dots,n\}$. Let $\Delta \mathcal{F}$ be $\{E\subseteq[n]\...
Tri's user avatar
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Let $G$ be a graph with $n$ vertices. Let $k$ be a positive integer. Assume that $G$ has no proper $\left(k-1\right)$-coloring. Then, the number of proper $k$-colorings of $G$ is at most $k! \cdot k^{...
darij grinberg's user avatar
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Given a constant $A$, among integrable functions $f: [0,1] \to \mathbb{R}^{+}$ such that $\int_0^1f(x)\mathrm{d}x= A$ and $f(0)=f(1)=0$, which function(s) maximise the probability $$\mathbb{P}(X_t \...
WRayn's user avatar
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Let $n$ be an integer, and consider an $n$ by $n$ grid of squares. We want to determine the least integer $k$ such that there exists a $k$-colouring of the squares so that there is no 2 by 2 subgrid ...
WRayn's user avatar
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2 votes
1 answer
399 views

I am trying to understand what the author means by $\mathcal{F}^N$ on the bottom of page 3 of the paper "Entropy approach for a generalization of Frankl's conjecture" by Veronica Phan, and ...
Fabius Wiesner's user avatar
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Inspired by this question, and in particular by the first part of it: Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$. ...
Fabius Wiesner's user avatar
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5 votes
1 answer
353 views

The recent resolution of the Kahn-Kalai conjecture (now Park-Pham theorem) used techniques developed in a slightly earlier breakthrough result by Alweiss, Lovett, Wu, and Zhang, which gives improved ...
Jarosław Błasiok's user avatar
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0 votes
0 answers
121 views

I am thinking of extending the Erdős–Szekeres theorem on monotone sequences. The original theorem (roughly) states: Among distinct $n$ real numbers, there is a monotone subsequence of length $\sqrt{n}...
Jaehyeon Seo's user avatar
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3 votes
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I would like to find an example of a lattice $L$ with the following properties: it has $q$ coatoms $x_i$, $1 \le i \le q$; for each coatom $x_i$ let $Y_i = \{y \in L :y \le x_i\} = (x_i]$: it is ...
Fabius Wiesner's user avatar
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2 votes
1 answer
246 views

Consider a finite lattice $L$ such that each atom has at least $|L|/2$ elements greater than or equal to it. It can be for example a boolean lattice or the following lattice: In the boolean lattice ...
Fabius Wiesner's user avatar
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1 vote
0 answers
127 views

Definition For graphs $H$ and $G$, a homomorphism from $H$ to $G$ is a function $f: V(H) \to V(G)$ such that $f(u)f(v) \in E(G)$ whenever $uv \in E(H)$. Let $\hom(H, G)$ denote the number of ...
West Book's user avatar
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1 vote
0 answers
98 views

Let $\mathcal{F}_1, \ldots, \mathcal{F}_h$ be families of finite sets. Let $\mathcal{G}_{i,j} = \mathcal{F}_i \cap \mathcal{F}_j, 1 \le i \lt j \le h$. We know that $|\mathcal{G}_{i,j}| \ge h$ and $\...
Fabius Wiesner's user avatar
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1 vote
0 answers
161 views

Given a function $f:{X\choose d}\to {X\choose \ell}$ where $X=\{1,2,\ldots,n\}$, $n>d>\ell$ and $(\forall A\in {X\choose d}) \ f(A)\subset A$. Prove that $$\log|Range(f)|=\Omega\left(\ell\log\...
Nader Bshouty's user avatar
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Let $G(A,B)$ be a bipartite graph with $|A| = |B| = n$, where $n$ is sufficiently large(thus, $o(n)/n,o(n^2)/n^2\ll 1$). The edge density of $G$ is $d = \frac{e(A,B)}{n^2}$, where $e(A,B)$ denotes the ...
West Book's user avatar
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4 votes
0 answers
382 views

[Now crossposted at math.stackexchange] Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is allowed to contain multiple instances of ...
Fabius Wiesner's user avatar
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2 votes
1 answer
199 views

Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is allowed to contain multiple instances of the same set. Let $U(\mathcal{F})$ be the ...
Fabius Wiesner's user avatar
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7 votes
4 answers
576 views

Say that an algebra of sets $K$ distinguishes set $B$ from set $C$ provided that for some $A\in K$, we have exactly one of $A\cap B$ and $A\cap C$ non-empty. Given families $F$ and $G$ of sets, say ...
Alexander Pruss's user avatar
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9 votes
3 answers
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Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
Fabius Wiesner's user avatar
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4 votes
0 answers
150 views

Consider a union-closed family $\mathcal{F}=\{A_1,…,A_n\}$ of $n \gt 1$ finite sets. I was not able to find a counterexample to the following conjecture: there exist two sets $A,B \in \mathcal{F}$ ...
Fabius Wiesner's user avatar
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4 votes
1 answer
462 views

[Now crossposted at math.stackexchange] Let $\mathcal{F} = \{\{x_1, x_2\} : 1 \le x_1 \lt x_2 \le n \}$, $n \ge 8$, and let $\mathcal{G} = \{G_1, \ldots, G_n\}$ be a partition of $\mathcal{F}$ in $n$ ...
Fabius Wiesner's user avatar
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0 votes
1 answer
165 views

I have a combinatorial question which is out of my research area. Given a $2^k\times 2^k$ matrix $A=[a_{i,j}]$ with entries in $\lbrace0,\pm1\rbrace$, where $k$ is a positive integer. Is it possible ...
Masayoshi Kaneda's user avatar
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1 vote
2 answers
498 views

Consider a family $\mathcal{G} = \{ A_1,B_1,\ldots,B_m \}$ of $m+1$ non-empty finite distinct sets with the following property: $$A_1 \cap B_k = \emptyset, 1 \le k \le m$$ Let $\mathcal{F} = \{A_1 \...
Fabius Wiesner's user avatar
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2 votes
1 answer
183 views

Imagine we have $r$ subsets of a ground set $S$, each of size $k$, such that each set of size $l$ is contained in at most one of the $r$ sets. What can we say about the minimum value of $|S|$? I am ...
Tuatarian's user avatar
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0 votes
0 answers
103 views

Say that we draw a graph in the following way: we first draw $n$ planar embeddings of $K_{2,2}$ (that is, we first draw $n$ quadrilaterals) such there are no edges which cross. Then for each of the $...
Avi's user avatar
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0 votes
1 answer
102 views

Let $M$ be a $m \times n$ matrix with binary entries (i.e. a matrix all whose entries belong to the set $\{0,1\}$), with $m\geq n$. Suppose each row of $M$ contains exactly $k$ ones. Given $n$ ...
Albert Garreta's user avatar
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0 votes
1 answer
312 views

I know that it is possible to represent every finite lattice $L$ with a union-closed family $\mathcal{F}$ containing the empty set: for every $x\in L$, let $S_x=\{y\in L\, :\, y\not\geq x\}$ and $\...
Fabius Wiesner's user avatar
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2 votes
1 answer
318 views

The Davenport constant $D(G)$ of a finite group $G$ is the minimal $d$ such that any sequence/multiset of length $d$ is one-product, i.e., identity can be obtained as a product (in some order) of some ...
Mikel Martinez Puente's user avatar
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1 vote
0 answers
227 views

[Crossposted at math.stackexchange]. Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$. I have written and solved ...
Fabius Wiesner's user avatar
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2 votes
0 answers
105 views

Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions $W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
West Book's user avatar
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3 votes
1 answer
432 views

[Crossposted at math.stackexchange.] Originally I posted a slightly more complicated version of this question. I decided to edit and put it in this simplified form, because I think that if we can ...
Fabius Wiesner's user avatar
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0 votes
0 answers
200 views

I am working on computing the Davenport constant $D(G)$ for $S_n$ and $A_n$, i.e., the minimal number $d$ such that every sequence (multiset) of $d$ elements contains some subsequence giving identity ...
Mikel Martinez Puente's user avatar
  • 117
3 votes
1 answer
372 views

I am working on computing the Davenport constant $D(G)$ of symmetric groups, which is the minimal number $d$ such that every sequence of $d$ elements, possibly with repetitions, is one-product, i.e. ...
Mikel Martinez Puente's user avatar
  • 117
3 votes
0 answers
198 views

I am working on the Davenport constant for symmetric groups, $D(G)$ , which is the minimal number $d$ such that every sequence of $d$ elements in the group G is one-product sequence, i.e, we can ...
Mikel Martinez Puente's user avatar
  • 117
1 vote
0 answers
191 views

Let $\mathbb{N}$ be the set of non-negative integers. Let $n\geq 2, d$ be positive integers. I would like a lower bound on the largest integer $r$ for which the following property holds: For any ...
Ben's user avatar
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10 votes
1 answer
1k views

[Now crossposted at math.stackexchange.] Consider a union-closed family $\mathcal{F}$ of $n$ finite sets with $\mathcal{F} \not = \{ \emptyset \}$. Let $\mathcal{H} \subseteq \mathcal{F}$ be the ...
Fabius Wiesner's user avatar
  • 1,186
3 votes
1 answer
160 views

Consider a union-closed family of sets $\mathcal{F}$, with $n = \vert\mathcal{F}\vert$ and thus $n \choose 2$ unordered couples of distinct sets $\{A, B\}$, $A,B \in \mathcal{F}$. In general, the ...
Fabius Wiesner's user avatar
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4 votes
2 answers
727 views

Version 2 of the conjecture was disproved. In this version 3 of the conjecture I am adding a further requirement to obtain from $\mathcal{H}$ a "minimal" family $\mathcal{G}$. I have already ...
Fabius Wiesner's user avatar
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