The extremal-combinatorics tag has no usage guidance.

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### Lower bound construction for Multidimensional Szemerédi's Theorem

The Multidimensional version of Szemerédi's theorem given by Theorem 10.2 in Tim Gower's paper from 2007 has the following statement.
Let $\delta>0$ and $k\in\mathbb{N}$. Then if $N$ is ...

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26 views

### Possible Number of Repetation of a Submatrix

Notation:
$H$ is the adjacency matrix of graph $H'$ respectively. $H_k$ is the block or sub-matrix of matrix $H$. The adjacency matrix of graph $H_k \cup H_e$ (subgraphs of $H'$) is $M_{(k,e)}$ ...

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**1**answer

59 views

### Reference for Turan Density

I am working a 3-graph problem. I convert it to calculate Turan density, that is $lim_{n\to \infty}\frac{ex_3(n,F)}{\binom{n}{3}}$, where F is a3-graph. I'd like to know are there some methods and ...

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votes

**1**answer

127 views

### Maximum size of minimal sequence of transpositions whose product is a given permutation

Consider the sequence $S = 1,2,3,\ldots n$ of elements, along with a sequence $T = t_1, t_2, \ldots, t_m$ of transpositions. Each transposition $t_i$ is a tuple $(a_i, b_i) \in [n]^2$. When applying a ...

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votes

**1**answer

156 views

### 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 ...

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59 views

### Tight bound of Turan number for K_{1,t,t}?

I'm looking for a tight bound for Turan number $ex_2(n,K_{1,t,t})$, where $K_{1,t,t}$ is the complete 3-partite graph with parts of size 1, t, and t.
The motivation is that we now $ex_2(n,K_{t,t})=O(n^...

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67 views

### Cases of equality in Daykin's theorem

Let $A$ and $B$ be sets of subsets of $\{1, \dots, n\}$, and let $A \wedge B = \{a \cap b : a\in A, b\in B\}$, $A\vee B=\{a\cup b: a\in A, b\in B\}$. Then
$$
|A \wedge B| |A\vee B| \geq |A||B|,
$$
as ...

**3**

votes

**1**answer

113 views

### An extremal combinatorics problem involving column summation

Given $n\in\Bbb N$, $\alpha>0$, $\beta\in\big[\frac12,1\big]$ denote $\mathcal R_{n,\alpha,\beta}$ as collection of all $2^n\times 2^{n^\alpha}$ $0/1$ matrices with every row summing to strictly ...

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votes

**1**answer

77 views

### Vertex cover of regular graph

(1.) How small can set $S$ of vertices in any regular undirected graph $G$ on $n$ vertices with degree $\Omega(n^\alpha)$ where $\alpha\in(0,1)$ can be such that every edge in the graph is incident on ...

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votes

**4**answers

233 views

### How small can a set system containing a large subset of every set be?

Fix $1>c>0$. Consider the set $[n]=\{1,2,\ldots,n\}$ and the set of all subsets of this set which we'll denote as $2^{[n]}$. Let $S \subseteq 2^{[n]}$ be a set system such that for every non-...

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votes

**2**answers

355 views

### 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,...

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31 views

### A class of unimodular parametrization

Is there a parametrization of set of matrices $\mathcal M\subseteq\Bbb Z[x_1,\dots,x_{m}]^{n\times n}$ such that $\forall f:\{-1,+1\}^{m}\rightarrow\{-1,+1\}$ $\exists M\in\mathcal M$ such that $\...

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55 views

### An extremal combinatorics problem

What is the minimum rank $r$ of an $n\times n$ square positive integer matrix such that sum of entries of every $\sqrt r\times\sqrt r$ submatrix is distinct and such that difference between minimum ...

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86 views

### 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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**0**answers

209 views

### 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 ...

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**1**answer

143 views

### Extremal combinatorics on bipartite graphs

One open question in extremal graph Theory is the so-called Zarankiewicz problem
(see for instance the wikipedia page), which ask for the maximum number of edges in a bipartite graph with a fixed ...

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**0**answers

170 views

### 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 ...

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1k views

### List of proofs where existence through probabilistic method has not been constructivised

Probabilistic method as first pioneered by Erdős (although others used this before) shows existence of a certain object. What are some of the most important objects for which we can show existence but ...

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83 views

### 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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**2**answers

280 views

### extremal bipartite graph

I'm facing the following question:
Given a bipartite graph $G = (L \cup R, E)$.
Let $n = |L|$, $m = |R|$, and a parameter $k \in \mathbb{N}$, $n > m > k$.
What is a minimal possible number ...

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**1**answer

214 views

### 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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votes

**1**answer

140 views

### 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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66 views

### 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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334 views

### What is the maximum size of a set system where the intersection of any two incomparable members is not in the set?

Let the set $\mathcal{F}$ consist of subsets of $[n]$. Suppose that for any incomparable $A$ and $B$ in $\mathcal{F}$, we have $A \cap B \notin \mathcal{F}$. What is the largest possible size of $\...

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267 views

### Maximal number of subsets in $\{1,\dots,n\}$ such that neither is contained in a union of two others

What are known estimates for maximal $M$ for which their exists subsets $A_1,\dots,A_M$ in $\{1,\dots,n\}$ such that there do not exist different indexes $i,j,k$ for which $A_i\subset A_j\cup A_k$?
...

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353 views

### Worst case difference in rank by column-row swapping

Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...

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votes

**1**answer

208 views

### Largest symmetric matrix given rank

Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$.
What is minimum ...

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72 views

### Minimum rank of certain matrices

Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in $\...

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65 views

### Euclidean minimum spanning trees intersecting each unit square

The recent question "Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell" can be restated in terms of "minimum spanning trees intersecting each (closed) lattice square of an $n\...

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votes

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### Random sequence of integers in $\{1, 2, \dots, n \}$ which is “everywhere probably increasing” - how long can it be?

Let $D=(d_1,d_2,\dots,d_k)$ be a sequence of correlated random variables. $D$ is "everywhere $r$-probably increasing" if the event $d_j > d_i$ has probability $\geq r$ for all $j > i$.
Fix $r \...

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**1**answer

129 views

### 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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votes

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228 views

### Number of subsum of a given set of integers

I am asking myself for few days the following but I can't find any references, although I am pretty sure this was already studied, because it seems quite natural:
Given n integers $(e_i)$ chosen (...

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**1**answer

242 views

### Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric).
Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix.
Case $1$: $M+W\in\{0,1\}^{n\times n}$.
Could ...

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vote

**1**answer

203 views

### Probabilistic statement on matrix ranks

Given $A\in\{0,1\}^{n\times n}$ with $\operatorname{rank}(A)=r=2^{O((\log_2n)^{\frac{1}{c+1}})}$.
Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s.
Does
$$\lim_{n\rightarrow\infty}\mathsf{P_{A\...

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88 views

### On subset of Deterministic games

Denote strings $u,v$ from $\{0,1\}^n$.
Denote concatenated pair $[uv]$.
Denote
$$[uv]_{1}=\{[uv]\oplus e_i\}_{i=1}^{2n}$$
collection of pairs with Hamming distance $1$ from $[uv]$ string ...

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votes

**1**answer

118 views

### Measuring how closely a missing projective plane can be approached by an equivalent structure

It is well known that for a number of structures, their existence is equivalent to the existence of a projective plane for a given order. Some of them depend on more than one parameters, which means ...

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### Families of Sets with Two Intersection Numbers

Let $k$ and $n$ be natural numbers. Let $I$ be a set of natural numbers. Let $\mathcal{F}$ be a family of $k$-element subsets of $\{ 1, \ldots, n\}$ such that $A, B \in \mathcal{F}$, $A \neq B$, ...

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**1**answer

906 views

### Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)

Is there a "Cauchy-Schwarz proof" of the following inequality?
Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) \,...

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votes

**1**answer

407 views

### 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 ...

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### combinatorial rectangles

Consider the set $S$ of all $m\times m$ matrices with $0-1$ entries with exactly $T$ combinatorial rectangles of all $0$s or all $1$s that partition each matrix in a non-overlapping manner.
Is there ...

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**2**answers

182 views

### 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 $b$ ...

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### open question on intersecting rectangles - reference request

In the now-inactive (and very nice) "Tricki" website that describes various proof techniques, there is an article (written I think by Timothy Gowers) that describes the following Ramsey-type problem:
...

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133 views

### Trees with a maximal convex hull: are the only optimal solutions Steiner trees?

For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take $\dfrac{...

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votes

**1**answer

524 views

### A method for making a graph bipartite

Given any graph $G$, can we find a bipartite subgraph of $G$ with at least $e(G)/2$ edges ($e(G)$ is the number of edges in $G$) by sequentially deleting the edge belonging to the most number of odd ...

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votes

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265 views

### A bound on a set

Let $x_1,\cdots , x_n$ be a sequence of real number such that $x_i\geq 1$ for all $1\leq i\leq n$, $S=\{\alpha_1x_1+\cdots +\alpha_nx_n | \alpha_i\in\{0,+1,-1\}\}$ and $I=[a,b)$ be a Interval with ...

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325 views

### 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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votes

**1**answer

115 views

### 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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votes

**3**answers

520 views

### Number of binary vectors of a given Hamming weight in a subspace of the Hypercube

Let $n$ be a natural number.
Let $U \subseteq \mathbb{F}_2^n$ be a linear subspace of dimension $k$.
What is the maximum number of vectors in $U$ of Hamming weight $\ell$?
The case I am specifically ...

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**0**answers

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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 ...

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votes

**1**answer

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### Hitting sets (aka covers aka transversals) of Steiner triple systems

Does there exist a constant $c$ so that the lines of every Steiner
triple system on $v$ points can be covered by $cv$ points?
That is if $D \in STS(v)$ with point set $T=\{1,2,\ldots,v\}$ then ...