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3
votes
1answer
123 views

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 ...
0
votes
0answers
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 $M_{(k,e)}$ ...
0
votes
1answer
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 ...
4
votes
1answer
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 ...
4
votes
1answer
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 ...
0
votes
0answers
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^...
4
votes
0answers
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
1answer
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 ...
0
votes
1answer
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 ...
3
votes
4answers
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-...
10
votes
2answers
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,...
0
votes
0answers
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 $\...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
4
votes
0answers
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 ...
2
votes
1answer
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 ...
2
votes
0answers
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 ...
25
votes
7answers
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 ...
4
votes
0answers
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 ...
6
votes
2answers
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 ...
6
votes
1answer
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 ...
2
votes
1answer
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 $...
3
votes
0answers
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 ...
12
votes
2answers
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 $\...
16
votes
1answer
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 $A_i\subset A_j\cup A_k$? ...
0
votes
2answers
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 ...
3
votes
1answer
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 ...
0
votes
0answers
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 $\...
2
votes
0answers
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\...
34
votes
2answers
1k views

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 \...
3
votes
1answer
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 ...
3
votes
2answers
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 (...
3
votes
1answer
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 ...
1
vote
1answer
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\...
2
votes
0answers
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 ...
6
votes
1answer
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 ...
5
votes
0answers
125 views

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 $A, B \in \mathcal{F}$, $A \neq B$, ...
14
votes
1answer
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) \,...
13
votes
1answer
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 ...
2
votes
0answers
180 views

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 ...
2
votes
2answers
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 $b$ ...
4
votes
2answers
161 views

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: ...
7
votes
2answers
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{...
13
votes
1answer
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 ...
5
votes
2answers
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 ...
12
votes
2answers
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 ...
0
votes
1answer
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 $E \subseteq X$ ...
4
votes
3answers
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 ...
19
votes
0answers
909 views

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 ...
2
votes
1answer
149 views

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 ...