Questions tagged [extremal-combinatorics]
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24
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How many positions of a tiling polygon can occur simultaneousy?
Let $T$ be a polygon which tiles the plane. For an instance of $T$ (mirrored or not), call the set of its translates a position of $T$.
My question:
How many different positions can occur in ...
3
votes
0
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183
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Matrices with only two different entries and maximal determinant
Define $\mathcal M_n$ as the set of all $n\times n$ matrices of full rank with each entry either 1 or $x$.
I am interested in how big the determinant of such a matrix can be. For this, we define in a ...
0
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0
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110
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Simpler lower bound for couples of disjoint sets
This is similar to a previous question, but simpler, I suppose.
Let $\mathcal{B}$ be the family of all subsets of $[n]=\{1,2,\ldots,n\} $ of size $2$. Let $\mathcal{F} = \{\mathcal{A}_1,\ldots,\...
55
votes
1
answer
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Intersecting family of triangulations
Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let $\...
26
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2
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What is the best lower bound for 3-sunflowers?
A collection of $t$ sets $A_i$ is called a t-sunflower if $A_i \cap A_j = Z $ for all $i \neq j$ for some fixed $Z$. A well-known conjecture of Erdos and Rado says that there is a constant $C_t$ such ...
26
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3
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What is the smallest size of a shape in which all fixed $n$-polyominos can fit?
Let $n$ be an integer and consider all fixed $n$-polyominos, i.e., without rotation or reflection. I am interested in finding a shape in which all polyominos can embed. (It is OK if multiple ...
14
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1
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616
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Minimal "sumset basis" in the discrete linear space $\mathbb F_2^n$
For a set $C\subseteq \mathbb F_2^n$, let $2C=C+C:=\{\alpha+\beta\colon \alpha,\beta\in C\}$.
I want to find $C$ of the smallest possible size such that $2C=\mathbb F_2^n$. Let $m(n)$ be the size of a ...
11
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2
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772
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Blocking sets in three dimensional finite affine spaces
What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line?
Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0,...
8
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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 ...
6
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2
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Conjecture about partitions of the powerset without the empty set
I would like to have some ideas about possibilities of proving or disproving the following conjecture:
For any partition $\mathcal{F}=\{\mathcal{A_1},\ldots,\mathcal{A_m} \}$ of the powerset without ...
5
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2
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433
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Minimum number of transpositions to make two multiset permutations equal
I think this problem should have a known solution, but I wasn't able to find any reference.
Consider a multiset of size $n \cdot m$: it has $n$ elements, and all element multiplicities equal to $m$.
...
5
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234
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A combinatorial problem
What is the largest $m\times m$ $0/1$ matrix of real rank $n$ with every square submatrix sized at least ${n^{\gamma}}\times{n^{\gamma}}$ distinct for some fixed $\gamma>0$?
Upper Bounds: Number ...
4
votes
1
answer
814
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Existence of triangle-free graphs for regular graphs of degree at most n/2
It is known that for triangle-free graphs, if they are $d$-regular, then $2d\leq n$, where $n$ is the number of vertices. In words, the degree is less than or equal to half the number of vertices (...
4
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1
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What is the minimum number of independent sets for a graph with fixed numbers of vertices and edges?
Fix integers $V$ and $E_{\text{max}}$, and consider graphs $G$ with $V$ vertices and at most $E_{\text{max}}$ edges. What is the best lower bound that one can give on the number of distinct ...
4
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3
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Maximal pairwise distance between $k$ permutations
How can k permutations on n-set be arranged to maximize minimal pairwise Kendall tau distance (i.e. number of discordant pairs) between them?
For two permutations this is obviously when the second ...
4
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2
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381
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An extremal combinatorics problem
Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ what is the largest $m$ we have such that in the interval $(p^\alpha,p-p^\alpha)$ there are $m$ integers $a_1<\dots<a_m$ ...
3
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Derandomization barriers in complexity theory applicable as barriers to constructive arguments replacing probabilistic method
The probabilistic method as first pioneered by Erdős (although others used this before) shows existence of a certain object while finding that object may take exponential time.
(1) Is there any ...
3
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0
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110
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Minimum number of couples of sets with non-empty intersection in a union closed family
Every union closed family $\mathcal{F}$, $\emptyset \notin \mathcal{F}$, with $|\mathcal{F}| = n$ sets, must have at least $\frac{2}{3}\binom{n}{2}$ unordered couples of sets with at least one element ...
3
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2
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Choosing subsets to cover larger sets
Consider the set $S=\{1,2,\ldots, n\}$, and let $a<b<n$. What is the minimum number $f(a,b)$ such that there exist $f(a,b)$ subsets of $S$ of size $a$ for which any subset of $S$ of size $b$ ...
2
votes
1
answer
193
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Number of members of a separating union-closed family whose universe has given cardinality
If I'm not wrong, it is easy to prove the following statement:
If $n \leq 4$ is a natural number, if $\mathcal{F}$ is a union-closed family of non-empty sets, if the universe of $\mathcal{F}$ (i.e. ...
1
vote
1
answer
140
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Schur complement and depermuting an algorithm for $\mathsf{determinant}\bmod2$
Let $$M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$$ be a matrix in $\mathbb F_2^{n\times n}$ where $A\in\mathbb F_2$ and $D\in\mathbb F_2^{(n-1)\times(n-1)}$ are square.
Through the determinant ...
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101
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Lower bound for couples of disjoint sets in some partitions of the power set
Originally posted on MathStackExchange but without answers.
Consider partitions $\mathcal{F}=\{\mathcal{A_1},\ldots,\mathcal{A_n} \}$ of the powerset without the empty set element $Q = \mathcal{P}([n])...
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Another conjecture about couples of disjoint two-element sets
Let $\{A_1,B_1\},\ldots,\{A_k,B_k\}$ be all the distinct unordered couples of subsets, with $A_i \cap B_i = \emptyset, 1 \le i \le k$, that can formed from a set $\{C_1,\ldots,C_q\}$, $q \le \binom{n}{...
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Number of $b$-separated Sidon sets with pairwise difference set intersection bounds
Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ call a set of $m$ integers $a_1<\dots<a_m$ in the interval $(p^\alpha,p-p^\alpha)$ to be $b$-separated Sidon if:
$a_i-a_j\...