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Questions tagged [extremal-combinatorics]

Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions.

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9 votes
1 answer
219 views

What is the smallest graph of treewidth $k$ having less edges than the $(k+1)$-clique?

Treewidth is a graph parameter measuring how close a graph is to being a tree. I am interested in what is the minimal number of edges required for a graph to have treewidth $k$. A natural family of ...
10 votes
2 answers
366 views

Algorithm to check whether a given set is Sidon

We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct. What algorithms do ...
4 votes
1 answer
189 views

Concrete version of KKL Theorem

The Kahn–Kalai–Linial (KKL) Theorem says that for any balanced Boolean function $f:\{−1,1\}^n→\{−1,1\}$ we have $\max_i {\bf Inf}_i(f) = \Omega\left(\frac{\log n}{n}\right)$. I am looking for a ...
3 votes
1 answer
227 views

Cover a graph with complete graphs

I want to find the smallest possible function $k(n,m)$ such that for any graph $G$ with $n$ vertices and $m$ edges, there exists $n$ vertex sets $S_1,S_2,...,S_n\subseteq V$ each with size $k(n,m)$ ...
2 votes
1 answer
88 views

Regularity Lemma for Multi-Relational Graphs?

Is there an analogous to Szemerédi regularity lemma in the setting, where I have multi relational graph i.e. I have $n$ nodes, but instead of having edges to be in $\{0,1\}$ i.e. there is an edge or ...
0 votes
1 answer
220 views

On structure of graphs with average degree equal to maximum average degree

For a simple graph $G$, the $\text{average-degree}(G)=|E(G)|/|V(G)|$ and the maximum average degree $\text{mad}(G)=\max\{\text{average-degree}(H)\colon H \text{ is a subgraph of } G\}$. If $\text{...
5 votes
1 answer
262 views

$\rho OPT + k$ approximation for bin packing (Unpublished result of David P. Williamson)

I am currently stuck on Exercise 5.12 in this book, which is apparently an unpublished result of David P. Williamson according to the book notes. The problem asks to use randomized rounding and first ...
5 votes
1 answer
235 views

Computational complexity of Turán-type problems

According to Turán's theorem (with $r=n/2$), any graph $G$ with $n$ vertices and at least $n(n-2)/2$ edges must contain a clique of size $n/2+1$. My question is: how hard is it to find this clique?$^*$...
2 votes
1 answer
98 views

"Parity testing set" for disjoint pairs of sets

I'd like a construction of the following description. Let $V$ be a set of $n$ elements. I'd like a collection $X$ of subsets of $V$ such that for any pair $(P,Q)$ of disjoint subsets of $V$, there ...
2 votes
0 answers
111 views

$k$-XOR collision free families

Given parameters $n,k\in \mathbb N^+$, I'm interested in finding a set of binary vectors $V_{n,k}=\{v_1,\ldots,v_n\}$ of length that satisfies: $\forall i: v_i\in\{0,1\}^{z_{n,k}}$. The bitwise xor ...
2 votes
2 answers
142 views

On the coloring number of small graphs with small cliques

Given a parameter $k$, and a graph $G$ with $O(k^2)$ vertices that has a maximum clique with $\le k$ vertices, I want to investigate the maximum number of colors $C(k)$ needed to properly color $G$, i....
4 votes
1 answer
301 views

Which (almost) balanced Boolean function has smallest "total" influence

The well known Kahn–Kalai–Linial (KKL) Theorem says that for any Boolean function $f\colon \{-1,1\}^n \xrightarrow{} \{-1,1\}$ $$ \max_{i \in [n]} \{\mathbf{Inf}_i[f] \} \geq \mathop{\bf Var}[f] \cdot ...
9 votes
0 answers
142 views

Graphs with minimal-size induced subgraphs

I consider undirected graphs $G = (V, E)$ for which I write $\text{n}(G) := |V|$ the number of vertices and $\text{m}(G) := |E|$ the number of edges. For $d \in \mathbb{N}$, I say that $G$ is $d$-...
8 votes
3 answers
640 views

Constant in Komlos conjecture

Given $n$ vectors $v_1,\dots,v_n\in\Bbb R^N$ with $\|v_i\|_2^2\leq1$ at every $i\in\{1,\dots,n\}$, Komlos conjecture states that, there is a $c\in\Bbb R$ (independent of $n,N$) such that at some $\...
4 votes
0 answers
98 views

How to construct an $(n,m,k)$ ``separating set''?

This problem is probably known under some other name, if anyone has seen it before, a reference will be great. Given $n,m,k$ (for $m,k\ll n$), a $(n,m,k)$ separating set is a set of $n$-sized binary ...
5 votes
0 answers
105 views

Constructing a large bit-vector set with the following property

I would like to construct a set $S\subseteq\{0,1\}^{2n}$ that satisfy the property: $$\forall x\neq y\in S\ \ \exists k\in [n]:\forall i,j\in[n], \sum_{t=k+i}^{2n}x_t\neq\sum_{t=k+j}^{2n}y_t$$ In ...
7 votes
2 answers
352 views

A Combinatorial Problem on Extremal Set Theory

Given a ground set $[n]$, under what condition of parameters $a,b,c$ does a family of subsets $\mathcal{F}\subseteq 2^{[n]}$ with the following property exist? (i) $\forall S\in \mathcal{F}$, $|S|=a$....
1 vote
1 answer
104 views

reference request- property of subset of rows in a matrix

I am interested in the following quantity. Suppose we are given a matrix $M\in \mathbb{F}_2^{m\times n}$ and a string $z\in \{0,1\}^n$. I am interested in finding the largest subset $S$ of rows in M ...
3 votes
0 answers
110 views

On number of disjoint sets with small stack depth in a set of permutations

Given k-distinct permutations $\sigma_1,\sigma_2,...,\sigma_k \in S_n$ where $k \leq 2^{\sqrt{n}}$ and $k >1$ (note that k is much smaller than number of possible permutations on [n]), What is ...
1 vote
2 answers
838 views

Upper bound for number of independent sets

What is the tightest upper bound known for the number of independent sets in a graph?
6 votes
0 answers
96 views

Perfect hashing family variation - injectivity on $r$ disjoint sets

We denote by $[t]$ the set $\{1,2,\ldots,t\}$. A $(n,k)$-perfect hashing family is a set of functions $H=\{h_i:[n]\to[k]\}$ such that for every set $S\subset [n], |S|\leq k$, there exists some $h_S \...
5 votes
0 answers
106 views

What is the smallest deterministic construction of an ordered perfect hashing family?

A $(n,k)$-perfect hashing family is a family of functions $H=\{h_i:[n]\to[k]\}$ such that for every set $S\subset [n], |S|\leq k$, there exists some $h_S \in H$ such that $H_S$ is injective on $S$. ...
2 votes
1 answer
573 views

Upperbound the order of P3-free partition of P4-free graphs

A graph $G$ is $P_k$-free if and only if $G$ does not have an induced subgraph isomorphic to a path of $k$ vertices. Thus, $P_2$-free graphs are exactly independent sets (or stable sets). $P_4$-free ...
6 votes
1 answer
269 views

Constructing a k-perfect permutations family

I'm looking for a set of permutations over $n$ elements $\mathcal{P}=\{P_1,P_2,...,P_r\}$ of minimal size such that for every ordered subset of size $k$, $S=<x_1,x_2,...,x_k>, (x_i \in [n])$, ...
4 votes
0 answers
104 views

Constructing a small (n,k)-Covering Matrices family

Let ${\cal A}$ be a set of $k\times n $ matrices over ${\mathbb F}_2$. We call ${\cal A}$ a (n,k)-covering, if for every subset of columns $I=(i_1,\ldots,i_k)\subseteq [n]$, there is a matrix $A \in {\...
3 votes
0 answers
183 views

Building a "balanced" universal set

A set of length $n$ binary vectors $\mathcal{U}=\{u_1,..,u_r\}$ is called $(n,k)$-universal if for all $S\subset [n], |S|=k$, $|\mathcal{U}_{|S}|=2^k$, i.e. for every subset of indices of size $k$, ...
5 votes
1 answer
241 views

Kruskal-Katona Theorem with Majority?

I am interested in the following problem which seems like an extension of the Kruskal-Katona Theorem. Let $A_k \subseteq \{0,1\}^n$ be a subset of the hypercube such that every element in $A$ has ...
8 votes
2 answers
291 views

Lower bound on the size of maximum interval induced subgraphs of an $n$-vertex graph $G$

Let $H$ be a maximum induced interval subgraph of a graph $G=(V,E)$. If $n=|V|$, then what is the smallest number of $V(H)$? The number is at most $3n/4$: consider a set of disjoint $4$-holes. Can ...