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gilkalai.wordpress.com | ||
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yufeizhao.wordpress.com
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| | | | | This post is adapted from my new expository survey Extremal regular graphs: independent sets and graph homomorphisms. The earliest result in extremal graph theory is usually credited to Mantel, who proved, in 1907, that a graph on $latex {n}$ vertices with no triangles contains at most $latex {n^2/4}$ edges, where the maximum is achieved for... | |
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dominiczypen.wordpress.com
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| | | | | Suppose you want to have a graph $latex G = (V,E)$ with chromatic number $latex \chi(G)$ equaling some value $latex k$, such that $latex G$ is minimal with this property. So you end up with a $latex k$-(vertex-)critical graph. It is easy to construct critical graphs by starting with some easy-to-verify example like $latex C_5$... | |
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anuragbishnoi.wordpress.com
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| | | | | The Ramsey number $latex R(s, t)$ is the smallest $latex n$ such that every graph on $latex \geq n$ vertices either contains a clique of size $latex s$ or an independent set of size $latex t$. Ramsey's theorem implies that these numbers always exist, and determining them (precisely or asymptotically) has been a major challenge... | |
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almostsuremath.com
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| | | I will give a proof of the measurable section theorem, also known as measurable selection. Given a complete probability space $latex {(\Omega,\mathcal F,{\mathbb P})}&fg=000000$, we denote the projection from $latex {\Omega\times{\mathbb R}}&fg=000000$ by $latex \displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\pi_\Omega\colon \Omega\times{\mathbb R}\rightarrow\Omega,\smallskip\\ &\displaystyle\pi_\Omega(\omega,t)=\omega. \end{array} &fg=000000$ By definition, if $latex {S\subseteq\Omega\times{\mathbb R}}&fg=000000$ then, for every $latex {\omega\in\pi_\Omega(S)}&fg=000000$, there... | ||
