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0
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0answers
122 views

On matrix rank inequality

Let $A$ be a $\{0,1\}$ square matrix. Let $J$ be all $1$ matrix. Let $\bar{A}=J-A$. Is it possible for $rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1$ and $rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1$ for some ...
8
votes
1answer
244 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) ...
0
votes
0answers
32 views

Comparing least degrees of certain polynomials

Let $\Bbb K$ be an infinite or a finite field with $\mathsf{char\mbox{ }}\Bbb K\neq 2$ and let $M\subsetneq\Bbb K[x_1,\dots,x_n]$ be the set of multilinear polynomials. Fix $S\subsetneq\{0,1\}^n$ and ...
6
votes
0answers
171 views

Minimal “basis” in $n$ dimensional unit cube

Let's $$ B^n=\{\bar\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)|\alpha_i\in \{0,1\}\};~~~~n=1,2,\ldots $$ and let's $$ C\subseteq B^n, $$ $$ S(C)=\{\bar\alpha\oplus\bar\beta\ | \bar\alpha,\bar\beta\in ...
1
vote
0answers
75 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
128 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
139 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
120 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 ...
13
votes
1answer
336 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
248 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 ...
0
votes
1answer
85 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
301 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 ...
13
votes
0answers
545 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
107 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 ...
1
vote
1answer
112 views

2-neighborhood of a simplex

Let $\Delta$ be an $n-1$-simplex in ${\mathbb R}^{n-1}$. For each vertex $v$ of $\Delta$ let $H_v$ be the hyperplane through $v$ and parallel to the opposite facet. By 2-neighborhood of a simplex I ...
5
votes
1answer
233 views

Extremal functions for tournaments

We are looking at directed graphs with no loops or parallel edges, but given two vertices $x$ and $y$, we allow the presence of both the edge $(x, y)$ and $(y, x)$. Thus, if $G$ is a directed graph ...
16
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1answer
637 views

Infinite desert with waterpoints

UPDATE: I created a simple web-application, that allows the user to move waterpoints around, then automatically calculates a maximum set of interior-disjoint squares between the points (the algorithm ...
0
votes
2answers
454 views

An extremal combinatorics problem over Finite Rings

Cross Posting from: http://math.stackexchange.com/questions/462016/a-combinatorics-problem-over-finite-rings Consider the set $S$ of all non-zero vectors over $\Bbb Z_{q}$ of length $r$ whose ...
5
votes
0answers
151 views

Diameter of subset sum graph

We have a finite set $X$, a weight function $w: X\rightarrow \mathbb{Z}^+$, and constants $k\leq c\in\mathbb{N}$. Let the weight $w(S)$ of a set $S\subseteq X$ be the sum of the weights of its ...
5
votes
4answers
252 views

Better bounds for exact-intersection Erdős–Ko–Rado system?

What is the largest possible number of subsets of a $4n$-element set $X$, such that each subset contains precisely $2n$ elements, and such that each of the pairwise intersections of the subsets has ...
9
votes
2answers
381 views

Largest number of k-arithmetic progressions without a (k+1)-arithmetic progression

Suppose $A \subseteq \{1,\dots,n\}$ does not contain any arithmetic progressions of length $k+1$. What is the largest number of $k$-term arithmetic progressions that $A$ can have? (one may also wish ...
1
vote
0answers
73 views

What are the minimal numbers of $(2\ 2)$-tetrahedra?

This time my question is related to Ramsey number   $R(4\ 4; 3) = 13$. DEFINITIONS:   Functions   $c : \binom X3\rightarrow \{0\ 1\}$   are called 2-colorings of triangles in ...
4
votes
2answers
110 views

$2$-colorings of triangles, resulting in $(2\ 2)$-colorings of all tetrahedra

DEFINITIONS:   Functions   $c : \binom X3\rightarrow \{0\ 1\}$   are called 2-colorings of triangles in   $X$.   The $4$-element subsets   $A\subseteq X$   are ...
3
votes
0answers
468 views

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 ...
35
votes
0answers
773 views

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 ...
1
vote
0answers
71 views

min max intersections of finte sets

Let $X=\{1,2,\cdots,n\}$ and $\mathcal{F} \subset 2^X$ so that $\mathcal{F}$ contains sets of given cardinality $k_1,k_2,\cdots, k_N$ (here $|\mathcal{F}| = N)$. Let $\delta(\mathcal{F})$ be the ...
2
votes
1answer
381 views

Popular elements in cross-intersecting families

Let $\mathcal{T}$ and $\mathcal{S}$ be two families of subsets of $[n]$ such that for all $T_i\in \mathcal{T}$ and $S_j\in \mathcal{S}$, $|T_i \cap S_j| \neq\emptyset$ $|T_i| , |S_j| \leq t = ...
1
vote
0answers
252 views

An Pure intriguing counting problem of index sets

Hi Guys. The problem here seems like a homework, but I think that it is not that easy.It comes from a theorem I recently proved.The content of the theorem is not important, the issue is that I have no ...
1
vote
1answer
208 views

Lower bound of the size of a collection of subsets with a intersecting property

The following question is a open question related to coding theory : What is the maximal size of a collection of $(\frac{n}{2} + 1)$-elements subsets of an n-element set such that each pair of subsets ...
1
vote
2answers
203 views

small sums of entries in submatrices - strange phenomenon

Suppose that $x \in \mathbb{R}^{n}$ is a vector of small positive fractions, i.e. $x_{i} \approx \frac{1}{n}$. The exact values are unknown. I form the matrix $M=diag(x)-\frac{xx^{T}}{2}$ which is a ...
6
votes
1answer
319 views

Upper bound for the size of a $k$-uniform $s$-wise $t$-intersecting set system

Given integers $n \geq k \geq t \geq 1$ and an integer $s$, let $m(n,k,s,t)$ denote the maximum size of a family $\mathcal F$ of $k$-subsets of $[n]$, i.e. $\mathcal F \subseteq \binom{[n]}{k}$, such ...