The Tur\'{a}n number and probabilistic combinatorics

TL;DR

This paper introduces the probabilistic method in combinatorics, providing an elegant, unconventional proof of a classical theorem related to Turán numbers, highlighting its beauty and potential for new insights.

Contribution

It offers a novel probabilistic proof of a well-known theorem on Turán numbers, not previously documented in literature.

Findings

Probabilistic method effectively proves classical combinatorial theorems.

Unconventional proof offers new perspective on Turán numbers.

Highlights elegance and potential of probabilistic techniques in combinatorics.

Abstract

In this short expository article, we describe a mathematical tool called the probabilistic method, and illustrate its elegance and beauty through proving a few well-known results. Particularly, we give an unconventional probabilistic proof of a classical theorem concerning the Tur\'{a}n number $T(n,k,l)$. Surprisingly, this proof cannot be found in existing literature.