# 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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### 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 ...
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 ...
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### 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 ...
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 $k(n,m)$ ...
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### 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 ...
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### 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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### 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 ...
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 $k$, $S=<x_1,x_2,...,x_k>, (x_i \in [n])$, ...