# Questions tagged [extremal-combinatorics]

The extremal-combinatorics tag has no usage guidance.

178
questions

1
vote

1
answer

174
views

### Reference for a topological result

I am reading the short paper due to Erdös and Bollobás "On a Ramsey-Turán type problem", where they obtain a lower bound for the number of edges on an $n$-graph without $K_4$ as a subgraph ...

0
votes

0
answers

116
views

### On nilpotent singular $\mathbb F_2^{n\times n}$ matrices

Let $M$ be a $0/1$ matrix over $\mathbb F_2^{n\times n}$ with determinant $0$.
The set of such singular matrices form a semigroup.
The set of nilpotent matrices of size $n\times n$ form a semigroup.
...

6
votes

1
answer

226
views

### A Sauer-Shelah-like lermma for prefix tree

I proved a variant of the Sauer-Shelah lemma and I was wondering if something like that is already known.
Let $S \subseteq \{0,1\}^n $. We say that a set of coordinates $K \subseteq [n]$ is shattered ...

0
votes

0
answers

280
views

### Union-closed family with a certain property

Now crossposted at math.stackexchange.
Consider a union-closed family $\mathcal{F} = \{A_1, \dotsc ,A_n\}$ of $n$ finite sets, $n$ odd, $n \ge 3$, $A_i \neq \emptyset$, $i=1,\dotsc,n$.
Let $r=\frac{n+...

1
vote

0
answers

54
views

### Lower bound for the minimum of the maximum frequency of an element - with restrictions

Consider a family $\mathcal{F}$ of non-empty sets, with
$n=|\mathcal{F}|$ sets, $q=\left|\cup\mathcal{F}\right|$ elements in the universe, and $q\le n/4$.
It is known that of the $\binom{n}{2}$ ways ...

4
votes

0
answers

68
views

### Bounding the Betti numbers of Čech complexes in Euclidean space

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \ge 2$. Then let $C=C(S)$ denote the union of closed balls of radius $1$ centered at points of $S$.
For $0 \le j \le d-1$, how large can the ...

3
votes

0
answers

98
views

### 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
votes

0
answers

67
views

### Harper's theorem on the general Hamming graph

Let $G$ be a graph, and for every $S \subseteq V$, let $N(S)$ denote the neighborhood of $S$ (i.e., the set of vertices that have neighbors in $S$). The vertex expansion of $G$ is
$$ \min_{S\subseteq ...

5
votes

1
answer

139
views

### Graphs without short cycles and with linear number of edges

Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a non-decreasing function and let $X_f$ be the class of graphs where every $n$-vertex graph $G$ is $(C_3, C_4, \ldots, C_{f(n)})$-free, i.e. $G$ contains ...

9
votes

0
answers

149
views

### Number of triangle-free graphs with prescribed number of edges

This question is posted from StackExchange since it received no answer there.
Let $f(n, e)$ be the number of triangle-free graphs on $n$ vertices and $e$ edges. From empirical evidence, I am motivated ...

3
votes

1
answer

101
views

### Bounds for $\mathrm{ex}(n,K_{2,\dots,2}^{(r)})$

$\DeclareMathOperator\ex{ex}$We write $K_{2,\dots,2}^{(r)}$ to denote the $r$-uniform hypergraph with vertex set $\{1,2\}\times\{1,\dots,r\}$ and hyperedge set $\{(1,1),(1,2)\}\times \{(2,1),(2,2)\} \...

6
votes

2
answers

367
views

### Coloring of a graph representing the power set

For a positive integer $n$, let $\mathcal{P}$ be the power set of $[n]$. Consider the graph $G$ with $\mathcal{P}$ as its vertex set, and, for $S_1,S_2 \in \mathcal{P}$, the edge $(S_1,S_2)$ exists ...

1
vote

0
answers

88
views

### cone structure of complement of hyperplanes

I want to know if in $\mathbb{R}^{m+3}$ we consider the following hyperplanes:
\begin{cases}
(1-g)y-\sum_{i\in I}x_i=0, & \text{if $I\subset\{1,\cdots,m+2\}$},|I|=g\\
gy-\sum_{i\in I}x_i+\...

1
vote

1
answer

105
views

### $n^2$-Grid $3n$-Coloring Game: Can we color a n-square grid with 3n colors s. t. we can't select n colors to get an histogram with $\Theta(n^2)$ area?

The coloring game is a game played between Alice and Bob.
There exists a grid of size $n \times n$, where $n$ is a strictly positive integer.
Each cell of the grid can be colored with a color that ...

4
votes

1
answer

145
views

### Extremal problems in additive combinatorics (over finite fields)

As you may know, there has been very recently a big breakthrough concerning upper bounds for the capset problem over $\mathbb{F}_3^n$ (and further generalizations to $\mathbb{F}_q^n$). I was wondering ...

25
votes

3
answers

823
views

### 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 ...

1
vote

1
answer

118
views

### Chromatic number or independence number of the generalized Kneser Graph

For positive integers $n,k$ and $s$, where $0\le s<k$ and $k \le n$, we define the generalized Kneser graph $K(n,k,s)$ as follows: The vertices of $K(n,k,s)$ are the $k$-subsets of $[2n]$, i.e., we ...

3
votes

1
answer

483
views

### Sum of $q$-binomial coefficients

Denote by $ \binom{n}{k}_q = \prod_{i=0}^{k-1} \frac{ q^{n-i} - 1 }{ q^{k-i} - 1 } $, $ k = 0, 1, \ldots, n $, the $ q $-binomial (Gaussian) coefficients. These numbers are symmetric, in the sense ...

5
votes

2
answers

350
views

### Counting intersections of hyperplanes

This is a dublicate from stackexchange:
Consider two families of hyperplanes $F_1$ and $F_2$ in $\mathbb{R}^d$ both containing $n$ hyperplanes. We have that for all $f \in F_1$ and $g \in F_2$ that $f$...

1
vote

0
answers

115
views

### Probability puzzle on partitions

Consider a set $U$ of size $n$ and let $\mathcal{S}$ be the set of all $(n/2)$-subsets of $U$ (assume $n$ is divisible by 4). Let $P$ be a partition of $\mathcal{S}$ into $k$ blocks $B_1,\dots,B_k$.
...

4
votes

0
answers

141
views

### Can resolution of the Kadison-Singer Problem provide progress on the Komlos Conjecture?

This is not a concrete question, just some thoughts.
The Komlos Conjecture is as follows-
There exists an absolute constant $C>0$, such that the following holds:
For all $d$ and any set of vectors ...

1
vote

0
answers

99
views

### Number of intersections that must occur in any partition of a given size

Let $\mathcal{S}$ be the set of all $n$-element subsets of $[2n]:=\{1,\dots,2n\}$.
Consider a partition $\mathcal{P}$ of $\mathcal{S}$ into $m$ blocks $P_1,\dots,P_m$, where all except at most one of ...

3
votes

0
answers

87
views

### Minimum number of partitions of a set such that the same pair must occur in a block in at least half of them

For positive integers $k$ and $n$, let ${S} = \{1,\dots,k\ n\}$. Consider $\ell \ge 3$ partitions $P_1,\dots,P_\ell$ of ${S}$, where each $P_i$ splits ${S}$ into $n$ blocks all of size $k$.
Question: ...

3
votes

2
answers

246
views

### Ramsey-Turán density function is well defined

Define
$$RT(n,K_l,f(n))=ex_l(n,f(n))=\max_G\{e(G): K_l \not\subset G, v(G)=n, \alpha(G)\leq f(n)\}$$
and the Ramsey-Turán density function $f_l:(0,1] \to \mathbb{R}$ as
$$f_l(\alpha)=\lim_{n\to \infty}...

0
votes

1
answer

123
views

### A general Turan-like question

Thinking of an edge as of a $2$-clique, it's natural to consider a slightly more general question than Turan considered in his celebrated theorem: given $r \le k \le n$, what is the maximal possible ...

2
votes

1
answer

167
views

### Maximizing and minimizing the number of positive product $k$-subsets of an $n$-set

The question is simple but require some definitions. I came across resolving a certain inequality. If there is no closed answer is there a related sequence describing the situation?
Let
$$S\ :=\ \{X=...

8
votes

2
answers

784
views

### a Littlewood–Offord-type problem concerning the "cubical lattice"

Fix even $n$ and consider the boolean function $f : \{0, 1\}^n \rightarrow \{0, 1\}$, $f : (x_0, \ldots , x_{n - 1}) \mapsto (x_0 \vee x_1) \wedge (x_2 \vee x_3) \wedge \cdots \wedge (x_{n - 2} \vee ...

3
votes

1
answer

169
views

### Products of Mersenne numbers as sums of real numbers

A Mersenne number is a number of the form $2^k-1$ for some $k \in \mathbb{N}$. Consider the set of $2^n-1$ products of Mersenne numbers
$$M_n=\left\{ \prod_{k\in S} (2^k-1) : S \subseteq [n], S\neq \...

7
votes

1
answer

474
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

0
answers

38
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 ...

4
votes

3
answers

504
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

1
answer

129
views

### 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 ...

3
votes

1
answer

172
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 $A$ ...

12
votes

1
answer

184
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

2
answers

116
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

1
answer

232
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

1
answer

164
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

0
answers

122
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

0
answers

160
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

0
answers

105
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

1
answer

90
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

0
answers

69
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

1
answer

272
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

0
answers

94
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

1
answer

106
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

1
answer

262
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

0
answers

68
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

1
answer

54
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

1
answer

73
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

0
answers

109
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\...