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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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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{...
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5 votes
1 answer
225 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?$^*$...
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5 votes
1 answer
237 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$\rho OPT + k$ approximation for bin packing (Unpublished result of David P. Williamson)

I am currently stuck on Exercise 5.12 in this book, which is apparently an unpublished result of David P. Williamson according to the book notes.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 ...
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2 votes
1 answer
90 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 ...
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2 votes
0 answers
105 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 ...
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2 votes
2 answers
129 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....
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4 votes
1 answer
224 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 ...
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9 votes
0 answers
109 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 $d$-...
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  • 7,350
4 votes
0 answers
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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 ...
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5 votes
0 answers
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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 ...
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  • 9,358
7 votes
2 answers
339 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) $\forall S\in \mathcal{F}$, $|S|=a$....
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  • 437
1 vote
1 answer
96 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 ...
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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 ...
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  • 31
8 votes
3 answers
617 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 $\...
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1 vote
2 answers
718 views

Upper bound for number of independent sets

What is the tightest upper bound known for the number of independent sets in a graph?
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  • 155
6 votes
0 answers
94 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 \...
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5 votes
0 answers
105 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 $S$. ...
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2 votes
1 answer
377 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 ...
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3 votes
0 answers
182 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 $k$, ...
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4 votes
0 answers
102 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 {\...
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  • 9,358
6 votes
1 answer
259 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 $k$, $S=<x_1,x_2,...,x_k>, (x_i \in [n])$, ...
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  • 9,358
5 votes
1 answer
223 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 ...
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  • 656
8 votes
2 answers
286 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 ...
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