Topics in Combinatorics lecture 1.0 -- welcome

What people said

Cites Ward Beullens · 2021-01 · Mathematics Stack Exchange · on “Smallest asymmetric relation such that each $k$ elements are dominated by a common element.”

How large does a non-empty set $X$ need to be if there exists an asymmetric relation $R \subset X^2$ such that for any $k$ elements $x_1,\dots,x_k$ of $X$ there exists a $x \in X$ such that $(x,x_i) \in R$ for all $i$ from $1$ to $k$ ? For the purpose of this question we say that a relation is asymmetric if $(x,y) \in R$ implies $(y,x) \not \in R$ . Example: for $k=1$ the answer is $|X| \ge 3$ , because the 'rock-paper-scissors' relation does the trick: $X = \{r,p,s\}$ and $R = \{(r,p),(p,s),(s,r)\}$ does the trick, and you can check that it is not possible with $|X| \le 2$ . For $k=2$ , I believe the answer is $|X| \ge 7$ . Context: I followed the `Topics in Combinatorics' lecture series by

Cites James Thorn · 2021-06 · Mathematics Stack Exchange · on “Bounds on the size of intersecting subsets of $\mathbb{Z_n}$”

Here is the problem: Let $\mathbb{Z_n}$ be the set of integers mod $n$ , and for a subset $A \subset \mathbb{Z_n}$ and for $x \in \mathbb{Z_n}$ , let $A+x$ denote the set { ${a+x | a \in A}$ }. Given two subsets $A,B \subset \mathbb{Z_n}$ prove there exists $x$ such that $|(A+x)\cap B| \geq |A||B|/n$ . Here is my attempt: First note that if either $A$ or $B$ are empty, then the inequality is satisfied. Therefore, it is safe to assume that both $A$ and $B$ are non-empty. If $A$ , $B$ are disjoint, then we can always find an $x$ such that $(A-x) \cap B \neq \emptyset$ . (Take some $a \in A$ and some $b \in B$ and let $x = b -a$ mod $n$ , this ensures the sets are not disjoint because $a+x$ is

Cites Nuvishramun · 2022-01 · Mathematics Stack Exchange · on “Intersection of Shifted Subsets of $\mathbb{Z}_n$”

This question is taken from the example sheet in the description of this video [this video] . Let $\mathbb{Z}_n$ be the set of integers mod $n$ , and for a subset $A \subseteq \mathbb{Z}_n$ and $x \in \mathbb{Z}_n$ , let $A+x = \{a+x \in \mathbb{Z}_n \ | \ a \in A\}$ . Let $A$ and $B$ be subsets of $\mathbb{Z}_n$ . Prove that there exists an $x \in \mathbb{Z}_n$ such that $|(A+x) \cap B| \geqslant |A||B|/n$ . Investigate the reverse inequality. (That is, try to find sets of given cardinality for which $\max_x |(A+x) \cap B|$ is as small as possible.) My question concerns the second part of this question. For the first part, I found that $$ \mathbb{E}\Bigl[|(A+x) \cap B|\Bigr] = \frac{|A||B|}{n},