Questions tagged [extremal-combinatorics]
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152
questions
0
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34 views
The linear embedding complexity of subsets of $0/1$ cube
We say $\pi$ is a subset of the $0/1$ integer points in $t$ dimensions represented by coordinates $(x_1,\dots,x_t)$ of complexity $\log^ct$ if there is an $A$ of $2^{\log^ct}\times2^{\log^ct}$ ...
7
votes
1answer
422 views
Packing equal-size disks in a unit disk
Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but ...
1
vote
0answers
18 views
Inequality between union-closed families of sets and corresponding upward-closed families
This question is about an inequality for union-closed families of sets related to Frankl's conjecture and a result by Reimer. It relates the union-closed families and corresponding upward-closed ...
2
votes
0answers
45 views
Does an $(x, bx)$-biregular graph always contain a $x$-regular bipartite subgraph?
I guess a discrete-mathematics-related question is still welcome in MO since I was new to the community and learned from this amazing past post. The following claim is a simplified and abstract form ...
1
vote
1answer
48 views
Schur Complement and depermuting an algorithm for determinant modulo $2$
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 ...
0
votes
0answers
28 views
On a $(k,l)$-monochromatic Hamming distance in $\mathbb F_2$?
A $(k,l)$-monochromatic edit on a matrix $M\in\mathbb F_2^{n\times n}$ is the operation
$$M+A$$
where $A\in\mathbb F_2^{n\times n}$ is of rank $1$ and number of $1$'s in $A$ is $kl$ and there are $l$ ...
3
votes
1answer
122 views
Maximum number of subsets in which people co-exist with their friends
Let $P = \{1,\dots,p\}$ be a set of people. Consider partitioning $P$ into two disjoint sets, $A$ (of cardinality $a$) and $A^c = P-A$. Let us index $A$ as $A = \{A_1,\dots,A_a\}$. Each person in $A$ ...
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0answers
75 views
Good source for understanding shifting techniques in combinatorics
There are many theorems in extremal combinatorics (e.g. the Erdos-Ko-Rado theorem, the Kruskal-Katona theorem) that can be proved via "shifting" arguments. See for example the following ...
12
votes
1answer
171 views
Ternary sequences satisfying $ x_i + y_i = 1 $ for some $ i $
Consider a set of strings $ {\mathcal S} \subset \{0, 1, 2\}^n $ satisfying the following two conditions: 1.) every string in $ {\mathcal S} $ has exactly $ k $ symbols from $ \{0, 1\} $ (i.e., $ \...
3
votes
2answers
105 views
For $n$ different sets whose union has size $n+1$ ,can you remove the same point from each set while retaining their difference
Let $(A_i)_{i=1}^{n}$ be $n$ different sets. Say $Z := \bigcup_{i=1}^{n}A_i$.
Q1: Is it true that if $|Z| \gt n$ then you can find $x \in Z$ such that the $(A_i-{x)}$ are still all different?
Q2: If $|...
6
votes
1answer
214 views
Permutation function based on subsets
We have some subsets $A_1,\dots,A_k$ of $A=\{1,2,\dots,n\}$. For each permutation $\sigma$ of $A$, define $f(\sigma) = \sum_{i=1}^k g(\sigma,A_i)$, where if the earliest element of $A_i$
in $\sigma$ ...
5
votes
1answer
145 views
Minimum number of independent pairs in a matroid
Given a matroid $M$ with ground set $E$ of size $2n$, suppose there exists $A\subseteq E$ of size $n$ such that both $A$ and $E\setminus A$ are independent. What is the minimum number of $B\subseteq E$...
5
votes
0answers
108 views
Exponential bound for very weak sunflowers?
Call $r$ sets diverse if for every $0\le i\le r$ there is an element contained in exactly $i$ of them.
A family of sets is r-diverse if any $r$ of its members are diverse.
Is there for every $r\ge 3$ ...
7
votes
0answers
141 views
Szemerédi's regularity lemma for binary operations
Szemerédi's regularity lemma is an approximate structure theorem for
all large graphs (symmetric binary relations). There are versions for
multicolored graphs and directed graphs. Is there an ...
6
votes
0answers
92 views
A question related to the union-closed sets conjecture
Let $f(n)$ denote the maximum possible cardinality of a collection $\mathcal F$ of nonempty sets which is closed under unions ($X,Y\in\mathcal F\implies X\cup Y\in\mathcal F$) and is such that no ...
2
votes
1answer
62 views
Computation of cyclic van der Waerden numbers
Van der Waerden's theorem gives us a finite number $W(k,r)$ defined as the smallest positive integer $N$ such that for any $n\geq N$, any $r$-coloring of $[n]=\{1,\dots,n\}$ admits a monochromatic $k$-...
3
votes
0answers
66 views
Boundary differences in two graphs
Let $\Gamma, \Xi$ be two graphs with the same set of vertices $V$ with $n$ elements. Assume $\Gamma$ is connected. Write $\Gamma\cup \Xi$ (or $\Gamma\cap \Xi$) for the graph whose set of edges is the ...
2
votes
0answers
172 views
Do sparse graphs contain a single regular pair?
An easy corollary of the Szemerédi Regularity Lemma is that dense graphs contain linear sized $\varepsilon$-regular bipartite subgraphs whose density is similar to that of the parent graph. As noted ...
4
votes
0answers
82 views
Maximal number of smallest circuits in a matroid
It is known (see here for example) that, in a simple graph of odd genus $g$ with $n$ vertices and $m$ edges, the number of cycles of lenght $g$ is at most $\frac{n(m-n+1)}{g}$.
Since this can be be ...
1
vote
1answer
85 views
Schur triples question
So I've been reading through On the number of monochromatic Schur triples by Datrovsky on finding the minimal number of Schur triples. This means you're trying to 2-colour the set of the smallest n ...
5
votes
1answer
246 views
Size of a family of sets of $k$-separated functions over $\{0,1,\ldots,n-1\}$
For $n\ge 1$ we write $[n]$ to denote the set $\{0,1,\ldots,n-1\}$. Let $2^{[n]}$ be the set of all functions from $[n]$ to $\{0,1\}$. Let $\mathcal{F}$ and $\mathcal{G}$ be two nonempty subsets of $2^...
1
vote
0answers
47 views
4-cycles vs eigenvalue information on quasi-random graphs
My (philosophical) question arises from reading the wonderful paper of Chung-Graham-Wilson where the authors introduces the notion of quasi-random graphs.
The main purpose of the paper is to show ...
0
votes
1answer
52 views
Maximal families of equal length intervals consist of equilateral triangles
My question is a follow up to How to find n points on a plane so that as many pair of points as possible have the same distance? -- see the conjecture at the bottom of this post.
Let $\ n\ $ be a ...
1
vote
1answer
70 views
What is the minimal possible size of a subset of this semigroup satisfying the following conditions?
Suppose $A$ is some set. Let's define a pair semigroup over $A$ as $P[A] = (A\times A \cup \{0\}, \circ)$, where the $\circ$ operation is defined by the following two identities:
$\forall a \in P[A]$ ...
2
votes
0answers
45 views
How many edges can be in an unbalanced bipartite graph of girth $>6$?
Let $G = (V, E)$ be a bipartite graph with $n, m$ nodes in its bipartition and girth (shortest cycle length) $>6$.
There is a simple counting argument called the Moore Bounds that gives
$$|E| = O\...
4
votes
1answer
152 views
Complementing the red and blue boolean cube?
Given a boolean $0/1$ cube in $n$ dimensions with $2^{n-1}$ red and $2^{n-1}$ blue points can we complement the cube (blue becomes red and vice versa) in $\operatorname{poly}(n)$ transformations?
...
1
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0answers
36 views
Partition complexity measure of the boolean cube?
Given $n$ points $p_1,\dots,p_n$ in $\{0,1\}^d$ my goal is to find $m$ index sets $\mathcal I_1,\dots,\mathcal I_m$ on the condition that each index set is a subset of $\{1,\dots,n\}$ on the ...
1
vote
0answers
81 views
Maximum number of ways of splitting a set of points with an hyperplane
Given a set $S$ of $n$ points in $\mathbb{R}^d$, let $D_S$ be the set
$\{\mathbf{v}=|\mathbf{u}-\mathbf{u'}|: \mathbf{u},\mathbf{u'}\in S\}$ (where $\forall i=1,2,\ldots, d$, $\mathbf{v}_i=|\mathbf{u}...
2
votes
0answers
182 views
The drawn diagonals divide the $N\times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$
Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N\times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N\times N$ board into $K$ regions. For each ...
1
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0answers
20 views
Do small subsets of $S_n$ subgroups cover almost all permutation configurations of $S_n$?
Given integer $m\in[1,n]$ fix a set $\mathcal T$ of permutations in $S_n$. Then there are subgroups $G_1,\dots,G_m$ of $S_n$ so that $\mathcal T$ is covered by cosets of $G_1,\dots,G_m$.
For ...
1
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0answers
91 views
Are almost all permutation configurations from $S_n$ covered by small subsets subgroups of $S_n$?
Given integer $m\in[1,n]$ fix a set $\mathcal T$ of permutations in $S_n$. Then there are subgroups $G_1,\dots,G_m$ of $S_n$ so that $\mathcal T$ is covered by cosets of $G_1,\dots,G_m$.
Do we have ...
33
votes
4answers
1k views
Dividing a cake between $n-1$, $n$, or $n+1$ guests
A housewife is waiting for guests and has prepared a cake. She doesn't know how many guests will come, but it will be $n-1$, $n$, or $n+1$.
What is the minimal number $f(n)$ of pieces the cake ...
5
votes
2answers
792 views
Cardinality of certain subsets in vector spaces over finite fields
Assume that you have an $n$-dimensional vector space over a finite field (therefore the number of elements in the vector space is finite) and $F$ is a subset of this vector space which contains $m$ ...
1
vote
0answers
59 views
Polytopes that can be efficiently described and efficiently covered by cubes or simplices?
Is there a bounded convex polytope $\mathcal P\subseteq\mathbb R^n$ with $m$ vertices, whose vertex vectors span $\mathbb R^n$ (so $m$ is $\Omega(n)$) and just $O(poly(\log n))$ half-plane ...
2
votes
0answers
63 views
Restricted Erdos-Ko-Rado theorem reference
Consider a family $\mathfrak{F}$ of $k$ element subsets of $\{1,2,..,n\}$, where $n\geq 2k$, such that any two members of $\mathfrak{F}$ have nonempty intersection. The Erdos-Ko-Rado theorem asserts ...
4
votes
0answers
139 views
Large finite subsets of Euclidean space with no isosceles (or approximately isosceles) triangles
Here's a question in combinatorial geometry which feels very much like other questions I'm familiar with but which I can't see how to get a hold of. I'll actually propose two different questions on ...
6
votes
1answer
225 views
An inequality for rearrangement-style sums
The following is a holdover from my math contest days that I never got around
to solve.
We will use the notation $\left[ k\right] $ for the set $\left\{
1,2,\ldots,k\right\} $ whenever $k$ is a ...
7
votes
1answer
379 views
A proper definition of connectivity for hypergraphs
For usual graphs on $n$ vertices, a edge-minimal connected graph is nothing but a spanning tree of this graph. It is well-known that any spanning tree has $n-1$ edges.
I would like to know whether ...
3
votes
1answer
107 views
The least number of edges to add to a tree that would force a certain number of edge-disjoint cycles
Let $c(n,k)$ be the least integer such that if $G$ is a simple graph on $n$ vertices with $n + c(n,k) - 1$ edges then $G$ has $k$ edge-disjoint cycles.
Clearly, $c(n, 1) = 1$ and it not very hard to ...
4
votes
0answers
88 views
Minimum number of balanced partitions
For any multiset $x_1,x_2,\ldots,x_{2n}$ of positive real numbers, a partition into two nonempty subsets $(A,B)$ is called "balanced" if $\text{sum}(A)\geq\text{sum}(B)-\max(B)$ and $\text{sum}(B)\geq\...
0
votes
0answers
96 views
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\...
4
votes
2answers
358 views
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$ ...
2
votes
0answers
50 views
Is the finite projective plane stable as an extremal set system?
Let $\Sigma$ be a set of $|\Sigma| = n$ subsets of the universe $[n]$, each of size $k$, with the property that any two of these subsets intersect on at most one element. It is easy to see that the ...
2
votes
1answer
119 views
Distinguishing points by sets of given size
The problem is:
Given a finite set $X$ with size $x$ and let $B$ denote a family of $k$-element subsets of $X$, called blocks. What is the smallest possible number $n$ of blocks such that every ...
3
votes
0answers
89 views
chromatic number of plane using Cairo pentagonal tiling
Scale the Cairo pentagonal tiling so the short side is of length 1. Then it is easy to colour the tiling with 8 colours, two parallel ribbons of four colours each, to establish that the chromatic ...
20
votes
0answers
375 views
Straight-line drawing of regular polyhedra
Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...
1
vote
2answers
161 views
Extremal density of a graph without a non-backtracking $2k$-cycle
The current best bound for the maximum possible density of an $n$-node graph with girth (shortest cycle length) $>2k$ is of the form
$$ex(n \ \mid \ C_{\le 2k}) = O(n^{1 + 1/k}),$$
while the ...
8
votes
1answer
261 views
A generalization of Erdős-Ko-Rado theorem
Is there any result known about the following generalization of the Erdős-Ko-Rado theorem?
Let $n, k, r, s$ be positive integers. We call a family $\mathcal{F}$ of $k$-element subsets of $\{1,\ldots, ...
2
votes
1answer
71 views
If we take a small subset of a sparse/uniform set system, how uniform can we arrange for the subset to be?
Fix a positive integer $N$ and a real $\epsilon>0$. I'll write $[N]$ for the set $\{1,\dots,N-1,N\}$, $\mathcal{P}X$ to denote the power set of a set $X$, and $\#X$ to denote the number of elements ...
1
vote
1answer
113 views
Number of distinct 3-sets formed as subsets of a 4-uniform set family
Let $\mathcal{F} \subseteq \binom{[n]}{4}$ be a collection of size-4 subsets of $\{1,\ldots,n\}$, i.e., a 4-uniform set family.
Let $\mathcal{F'}$ be the collection of size-3 sets for which a ...
