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### Additional condition to the Bollobas theorem (Sperner's therorem) in extremal set theory

The Bollobas'1965 theorem is the following:
If $A_1,...,A_n$ and $B_1,...,B_n$ are two sequences of subsets of $X=\{1,...,r\}$ such that $A_i\cap B_j = \emptyset$ if and only if $i=j$, then
$$\sum_{i=...

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### Kruskal-Katona for multisets?

Following Fedor Petrov's remarks, here is a "set-theoretic version" of the
question I asked a while ago.
For integer $n\ge 1$, denote by $\mathcal M_n$ the family of all (finite)
multisets with ...

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### Kruskal-Katona for homocyclic groups?

I need a version of the Kruskal-Katona theorem (better still, of the Lovasz "approximate" version thereof) for the elementary abelian / homocyclic groups, in the following spirit:
What is the ...

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### Given k, what is the minimum n such that n choose n/2 is greater than k? [closed]

I'm not an expert in combinatorics, but it sometimes comes up in my research with students in computer science (which is already pretty far away from my speciality of abstract homotopy theory). I just ...

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### Families of subsets with pairwise symmetric differences of cardinality at most $k$

Let $X$ be an $n$-element set and $\mathcal{F} \subseteq P(X)$ such that for all $A, B \in \mathcal{F}$, $|A△B| \leq k$ where $A△B$ denotes the symmetric difference of $A$ and $B$. Suppose $|\mathcal{...

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### Combinatorics-the maximum number of subsets with a given property

Let $X$ be a set with $n$ elements. I would like to know the maximum number of subsets of $X$ such that the number of elements in the symmetric difference between any two of these subsets is at most $...

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### What is the maximal number of partitions with this maximal intersection property?

Let $X = \{ 1, \dots, n = sk \}$ be a finite set. Let $\mathscr P, \mathscr Q$ be equi-partitions of $X$ into $k$ sets of size $s$. Denote by $V(\mathscr P, \mathscr Q)$ the maximum size of ...

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### Minimal family of k-sets containing all t-sets

Let $n \ge k \ge t \in \mathbb{N}$, and consider a universe $U$ of size $n$. Let $\mathcal{F}$ be a family of $k$-subsets of $U$, such that every $t$-subset of $U$ is contained in at least one member ...

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### Set system with different differences

What is the maximal number of sets in a set system $\mathcal{A}$ of subsets of an $n$ element set such that for every $i \neq j $ and $A_i,A_j \in \mathcal{A}$ the difference $A_i \setminus A_j$ is ...

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### Generalized Helly theorem for $t$-intersecting families

Given a family $\mathcal{F}$ of sets over ground set $X$, let $\tau(\mathcal{F})$ be the transversal number (aka blocking number), that is the cardinality of the smallest set of points $E \subseteq X$ ...

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### Can a partition free family in $2^{[n]}$ always be enlarged to one of size $2^{n-1}$?

Let $\left[ n \right]=\{{1,2,\cdots,n\}}$ and call a family $\mathcal{F} \subset 2^{\left[n\right]}$ partition-free if it does not contain any partition of $\left[n\right]$. A recent question asked ...

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### The state of art of the sunflower lemma

I am interesting in the sunflower system and its applications in computer science.
Given a Universe $U$ and a collection of $k$ sets $A_i$ is called a k-sunflower system if $A_i \cap A_j = Y $ for ...

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### Optimal bounds for an alternating sum on a downset

Let $n$ be a natural number, and consider the discrete cube $2^{[n]} := \{ A: A \subset \{1,\ldots,n\}\}$ consisting of all subsets of the $n$-element set $[n] := \{1,\ldots,n\}$. Define a downset in ...

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### Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one.
Are there simple formulas if one ...

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### Intersecting Hamming spheres: is $|A\stackrel k+E|\ge|A|$?

Since my original posting some ten days ago, I discovered an amazing
example which changed significantly my perception of the problem.
Accordingly, the whole post got re-written now.
The most ...

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### Minimal generation for finite abelian groups

Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:
1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,
2) With orders that are ...

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### The hypercube: $|A {\stackrel2+} E| \ge |A|$?

I have a good motivation to ask the question below, but since the post is
already a little long, and the problem looks rather natural and appealing
(well, to me, at least), I'd rather go straight to ...

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### Lower bounding the maximum size of sets in a set family with union promise

The following problem has come up while working on the relationship between certificate and randomized decision tree complexities of boolean functions. However, I think it is of interest by itself and ...

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### How to keep subsets disjoint?

Given positive integers $n$ and $k\le 2^n$, how to choose a subset $C\subset\{0,1\}^n$ of size $|C|=k$ to maximize the number of pairs $(c_1,c_2)\in C\times C$ with the supports of $c_1$ and $c_2$ ...

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### What is the largest family F of subsets of [n] for which any two distinct sets A and B in F have an intersection of size at most min(|A|,|B|)/2?

This problem arose in the study of Latin squares with a large number of subsquares, although it appears interesting in its own right.
Question: What is the maximum cardinality of a family $F \...

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### Upper bound for the size of a $k$-uniform $s$-wise $t$-intersecting set system

Given integers $n \geq k \geq t \geq 1$ and an integer $s$, let $m(n,k,s,t)$ denote the maximum size of a family $\mathcal F$ of $k$-subsets of $[n]$, i.e. $\mathcal F \subseteq \binom{[n]}{k}$, such ...