# Combinatorics

Published articles for Combinatorics.

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## Socks, a matching game based on an additive combinatorics problem

DevFeed: [Socks, a matching game based on an additive combinatorics problem](<https://devfeed.tech/articles/socks-a-matching-game-based-on-an-additive-combinatorics-problem-40478.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2023/10/14/socks-a-matching-game-based-on-an-additive-combinatorics-problem/>)

Published: 2023-10-14T06:00:00Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [math](<https://devfeed.tech/topics/math.md>), [structure](<https://devfeed.tech/topics/structure.md>)

Tags: [additive-combinatorics](<https://devfeed.tech/tags/additive-combinatorics.md>), [board-games](<https://devfeed.tech/tags/board-games.md>), [card-game](<https://devfeed.tech/tags/card-game.md>), [combinatorics](<https://devfeed.tech/tags/combinatorics.md>), [game](<https://devfeed.tech/tags/game.md>), [games](<https://devfeed.tech/tags/games.md>), [group-theory](<https://devfeed.tech/tags/group-theory.md>), [math](<https://devfeed.tech/tags/math.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [structure](<https://devfeed.tech/tags/structure.md>), [xor](<https://devfeed.tech/tags/xor.md>)

### AI overview

This article introduces Socks, a matching game modeled using six-dimensional binary vectors and additive combinatorics. It proves that any seven cards contain a valid zero-summing set, while six specific cards can avoid one, so the minimum guarantee is seven cards.

### Source excerpt

Can you find a set of cards among these six, such that the socks on the chosen cards can be grouped into matching pairs? (Duplicate pairs of the same sock are OK) Spoilers: If the cards are indexed as 1 2 3 4 5 6 Then the following three subsets work: $\{ 1, 2, 4, 5, 6 \}$, $\{ 2, 3, 6 \}$, and $\{ 1, 3, 4, 5 \}$.

## Optimization Models for Subset Cover

DevFeed: [Optimization Models for Subset Cover](<https://devfeed.tech/articles/optimization-models-for-subset-cover-40441.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2020/10/20/optimization-models-for-subset-cover/>)

Published: 2020-10-20T09:00:00Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Optimization](<https://devfeed.tech/topics/optimization.md>), [Code](<https://devfeed.tech/topics/code.md>), [Open Source](<https://devfeed.tech/topics/open-source.md>)

Tags: [article](<https://devfeed.tech/tags/article.md>), [boolean-satisfiability](<https://devfeed.tech/tags/boolean-satisfiability.md>), [combinatorics](<https://devfeed.tech/tags/combinatorics.md>), [integer-programming](<https://devfeed.tech/tags/integer-programming.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [optimization](<https://devfeed.tech/tags/optimization.md>), [programming](<https://devfeed.tech/tags/programming.md>), [python](<https://devfeed.tech/tags/python.md>), [smt](<https://devfeed.tech/tags/smt.md>), [source](<https://devfeed.tech/tags/source.md>), [subset-cover](<https://devfeed.tech/tags/subset-cover.md>), [z3](<https://devfeed.tech/tags/z3.md>)

### AI overview

This article explores subset covering as a combinatorial optimization problem and compares Integer Linear Programming solvers with Z3, an open-source SMT solver from Microsoft. It defines a generalized formulation for covering all subsets of size l with the fewest subsets of size k.

### Source excerpt

In a recent newsletter article I complained about how researchers mislead about the applicability of their work. I gave SAT solvers as an example. People provided interesting examples in response, but what was new to me was the concept of SMT (Satisfiability Modulo Theories), an extension to SAT. SMT seems to have more practical uses than vanilla SAT (see the newsletter for details). I wanted to take some time to explore SMT solvers, and I landed on Z3, an open-source SMT solver from Microsoft.

## The Erdős-Rényi Random Graph

DevFeed: [The Erdős-Rényi Random Graph](<https://devfeed.tech/articles/the-erdos-renyi-random-graph-40329.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/08/22/the-erdos-renyi-random-graph/>)

Published: 2013-08-22T16:11:18Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Graphs](<https://devfeed.tech/topics/graphs.md>), [graph theory](<https://devfeed.tech/topics/graph-theory.md>), [math](<https://devfeed.tech/topics/math.md>), [Networks](<https://devfeed.tech/topics/networks.md>)

Tags: [color](<https://devfeed.tech/tags/color.md>), [combinatorics](<https://devfeed.tech/tags/combinatorics.md>), [erdos](<https://devfeed.tech/tags/erdos.md>), [giant-component](<https://devfeed.tech/tags/giant-component.md>), [graph](<https://devfeed.tech/tags/graph.md>), [graph-coloring](<https://devfeed.tech/tags/graph-coloring.md>), [graph-theory](<https://devfeed.tech/tags/graph-theory.md>), [properties](<https://devfeed.tech/tags/properties.md>), [random-graph](<https://devfeed.tech/tags/random-graph.md>), [random-graphs](<https://devfeed.tech/tags/random-graphs.md>), [renyi](<https://devfeed.tech/tags/renyi.md>), [social-networks](<https://devfeed.tech/tags/social-networks.md>), [structure](<https://devfeed.tech/tags/structure.md>)

### AI overview

This tutorial introduces the Erdős-Rényi random graph model G(n,p), in which each possible edge between n vertices is included independently with probability p. It discusses the model's role in combinatorics and network analysis, along with basic properties, applications, and computational exploration.

### Source excerpt

During the 1950's the famous mathematician Paul Erdős and Alfred Rényi put forth the concept of a random graph and in the subsequent years of study transformed the world of combinatorics. The random graph is the perfect example of a good mathematical definition: it's simple, has surprisingly intricate structure, and yields many applications. In this post we'll explore basic facts about random graphs, slowly detail a proof on their applications to graph theory, and explore their more interesting properties computationally (a prelude to proofs about their structure).

## Ramsey Number Lower Bound

DevFeed: [Ramsey Number Lower Bound](<https://devfeed.tech/articles/ramsey-number-lower-bound-40293.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2012/12/02/ramsey-number-lower-bound/>)

Published: 2012-12-02T15:17:26Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [graph theory](<https://devfeed.tech/topics/graph-theory.md>), [Graphs](<https://devfeed.tech/topics/graphs.md>), [Mathematics](<https://devfeed.tech/topics/mathematics.md>)

Tags: [combinatorics](<https://devfeed.tech/tags/combinatorics.md>), [computational-complexity](<https://devfeed.tech/tags/computational-complexity.md>), [graph](<https://devfeed.tech/tags/graph.md>), [graph-coloring](<https://devfeed.tech/tags/graph-coloring.md>), [graph-theory](<https://devfeed.tech/tags/graph-theory.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [number-theory](<https://devfeed.tech/tags/number-theory.md>), [probabilistic-method](<https://devfeed.tech/tags/probabilistic-method.md>), [ramsey-numbers](<https://devfeed.tech/tags/ramsey-numbers.md>), [random-graph](<https://devfeed.tech/tags/random-graph.md>)

### AI overview

The article proves a lower bound for the diagonal Ramsey number R(m,m). By randomly coloring the edges of a complete graph, it shows that if binom(n,m)2^(1−binom(m,2)) is less than 1, there exists a coloring with no monochromatic m-vertex subgraph, so R(m,m) > n.

### Source excerpt

Define the Ramsey number $ R(k,m)$ to be the minimum number $ n$ of vertices required of the complete graph $ K_n$ so that for any two-coloring (red, blue) of the edges of $ K_n$ one of two things will happen: There is a red $ k$-clique; that is, a complete subgraph of $ k$ vertices for which all edges are red. There is a blue $ m$-clique. It is known that these numbers are always finite, but it is very difficult to compute them exactly.

## N Choose 2 is the Sum of the First N-1 Integers

DevFeed: [N Choose 2 is the Sum of the First N-1 Integers](<https://devfeed.tech/articles/n-choose-2-is-the-sum-of-the-first-n-1-integers-40243.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2011/10/02/n-choose-2/>)

Published: 2011-10-02T16:16:47Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>)

Tags: [arithmetic](<https://devfeed.tech/tags/arithmetic.md>), [bijections](<https://devfeed.tech/tags/bijections.md>), [combinatorics](<https://devfeed.tech/tags/combinatorics.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [proofs-without-words](<https://devfeed.tech/tags/proofs-without-words.md>)

### AI overview

A combinatorial bijection shows that the binomial coefficient n choose 2 equals the sum of the first n−1 integers. The article explains how yellow dots correspond uniquely to pairs of dots in the bottom row, then briefly connects bijections to isomorphism and classification in mathematics.

### Source excerpt

Problem: Determine an arithmetic expression for $ \binom{n}{2}$. Solution: The following picture describes a bijection between the set of yellow dots and the set of pairs of purple dots: In particular, selecting any yellow dots and travelling downward along diagonals gives a unique pair of blue dots. Conversely, picking any pair of blue dots gives a unique yellow dot which is the meeting point (the "peak") of the inward diagonals. If we say the bottom row has $ n$ elements, then the number of yellow dots is clearly $ 1 + 2 + \dots + (n-1)$, and the number of pairs in the last row is just $ \binom{n}{2}$.