Monthly Archives: May 2015

Notorious bound for Ramsey number for two coloring

Lemma. If r\ge2, then R\left(r,2\right)=r.

Proof. 꼭짓점이 r인 완전그래프 K_{r}을 생각해보자. 그러면 K_{r}에는 blue로 칠해진 edge가 적어도 한 개가 있거나(K_{2}) 모두 red인 경우거나 둘 중 하나다. 그러므로 R\left(r,2\right)\le r이다.

모두 red로 칠해진 완전그래프 K_{r-1}를 생각해보면 induced subgraph로서 blue로 된 K_{2}는 없고, red로 된 K_{r} 또한 없다. 그러므로 r-1<R\left(r,2\right)이므로 R\left(r,2\right)=r이다.


 

Theorem. If r\ge2 and s\ge2, then

(1)   \begin{equation*} R\left(r,s\right)\le R\left(r-1,s\right)+R\left(r,s-1\right). \end{equation*}

Also, we get a notorious bound

(2)   \begin{equation*} R\left(r,s\right)\le\binom{r+s-2}{r-1}. \end{equation*}

Proof. (1)과 (2)가 동시에 성립한다는 것을 보이기 위해, 그렇지 않는다고 하자. 다시 말해, (1)이 성립하지 않는 r,s>2
존재한다고 하자. 그리고 이런 r,s들 중에서 r+s가 최소가 되도록 잡자.

이제 u=r-1, v=s-1이라 하면, r+s의 최소성에 의하여 R\left(u,s\right),R\left(r,v\right),R\left(u,v\right)는 (1)을 만족해야 한다.

n=R\left(u,s\right)+R\left(r,v\right)라 하자. 이제 K_{n}의 red-blue coloring을 생각하고, v\in V\left(G\right)를 하나 생각해보자. 그리고 v와 연결된 점들을 색깔을 기준으로 분류를 해보자.

    \begin{align*} N_{R} & =\left\{ w\in V:w\ne v,\,\chi\left(\left\{ v,w\right\} \right)=\text{red}\right\} ,\\ N_{B} & =\left\{ w\in V:w\neq v,\,\chi\left(\left\{ v,w\right\} \right)=\text{blue}\right\} \end{align*}

라 정의하자. 그러면 N_{R}\cup N_{B}\cup\left\{ v\right\} =V\left(G\right)이므로, \left|N_{R}\cup N_{B}\right|=\left|N_{R}\right|+\left|N_{B}\right|-1=n-1이다. 그러므로 \left|N_{R}\right|\ge R\left(u,s\right)이거나 \left|N_{B}\right|\ge R\left(r,v\right) 둘 중 하나가 성립해야 한다. 이 사실은 자명하지 않은 사실이므로 증명이 필요하다.

그렇지 않다고 하자. 즉 다시 말해서 \left|N_{R}\right|<R\left(u,s\right)이고 \left|N_{B}\right|<R\left(r,v\right)라 하자. 그런데 램지수는 정수이므로

    \begin{align*} n-1 & =\left|N_{R}\right|+\left|N_{B}\right|\le R\left(u,s\right)-1+R\left(r,v\right)-1\\ & =R\left(u,s\right)+R\left(r,v\right)-2\\ & =n-2 \end{align*}

가 성립해야 하는데, 이는 모순이다.

\left|N_{R}\right|\ge R\left(u,s\right)일 때, 꼭짓점이 N_{R}인 완전그래프를 생각해보자. 그런데, \left|N_{R}\right|\ge R\left(u,s\right)이므로 N_{R}에는 적어도 빨간색으로 색칠된 완전그래프 K_{u}를 포함하고 있더나 또는 파란색으로 색칠된 완전그래프 K_{s}를 포함하고 있다. 만약 v가 처음에 언급한 완전그래프에 포함된다고 가정하면, N_{R}의 정의에 의하여 이는 빨간색으로 색칠된 완전그래프 K_{u+1}=K_{r} 를 갖거나, 파란색으로 색칠된 완전그래프 K_{s}를 가질 것이다.

마찬가지의 원리로 \left|N_{B}\right|\ge R\left(r,v\right)라 하면 N_{B}을 꼭짓점으로 갖는 완전그래프를 생각하자. 그 후 여기에 v를 하나 더 추가해서 완전그래프를 생각해보면 빨간색으로 색칠된 완전그래프 K_{r}을 가지고 있거나, 파란색으로 색칠된 완전그래프 K_{v+1}=K_{s}를 가질 것이다.

그러므로 위 관찰로부터 K_{n}의 색칠된 변에는 빨간색으로 색칠된 K_{r}을 갖거나, 파란색으로 색칠된 K_{s}를 가질 것이다. 이로부터 R\left(r,s\right)\le n이다.

그러므로

    \begin{align*} R\left(r,s\right) & \le n=R\left(u,s\right)+R\left(r,v\right)\\ & \le\binom{u+s-2}{u-1}+\binom{r+v-2}{r-1}\\ & =\binom{r+s-3}{r-2}+\binom{r+s-3}{r-1}\\ & =\binom{r+s-2}{r-1} \end{align*}

이므로 이는 처음에 (1)와 (2)가 성립하지 않는다는 사실에 모순이다.

Lp norm and distribution function with dyadic series

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Proposition. For any measurable function f:X\rightarrow\mathbb{C} and 0<p<\infty we have

    \[ \norm fp^{p}\sim_{p}\sum_{n\in\mathbb{Z}}\lambda_{f}\left(2^{n}\right)2^{np}. \]

Here X\sim_{p}Y means there are constants c_{p},C_{p}>0 depending only on p so that c_{p}Y\le X\le C_{p}Y.

Proof. For each n\in\mathbb{Z}, define

    \[ E_{n}=\left\{ x\in X:\left|f\left(x\right)\right|\ge2^{n}\right\} . \]

Then \lambda\left(2^{n}\right)=\mu\left(E_{n}\right) for all n\in\mathbb{Z}. Also, E_{n}\supset E_{n+1} for all n. Define A_{n}=E_{n}\setminus E_{n+1}. Then \bigcup A_{n}=\bigcup E_{n}=X and \left\{ A_{n}\right\} is a family of disjoint sets in X. Then we have

    \begin{align*} \norm fp^{p} & =\int_{X}\left|f\right|^{p}d\mu\\  & =\int_{\bigcup_{n\in\mathbb{Z}}A_{n}}\left|f\right|^{p}d\mu\\  & =\sum_{n\in\mathbb{Z}}\int_{A_{n}}\left|f\right|^{p}d\mu. \end{align*}

Note that x\in A_{n} if and only if 2^{n}\le\left|f\left(x\right)\right|<2^{n+1} for all n\in\mathbb{Z}.

For the upper bound, we have

    \begin{align*} \norm fp^{p} & \le\sum_{n\in\mathbb{Z}}\left(2^{n+1}\right)^{p}\int_{A_{n}}d\mu\\  & =\sum_{n\in\mathbb{Z}}\left(2^{n+1}\right)^{p}\mu\left(A_{n}\right)\\  & =\sum_{n\in\mathbb{Z}}\left(2^{n+1}\right)^{p}\left(\lambda_{f}\left(2^{n}\right)-\lambda_{f}\left(2^{n+1}\right)\right)\\  & \le\sum_{n\in\mathbb{Z}}\left(2^{n+1}\right)^{p}\lambda_{f}\left(2^{n}\right)\\  & \le C_{p}\sum_{n\in\mathbb{Z}}\lambda_{f}\left(2^{n}\right)2^{np} \end{align*}

where C_{p}=2^{p}.

For the lower bound, we have

    \begin{align*} \norm fp^{p} & =\sum_{n\in\mathbb{Z}}\int_{A_{n}}\left|f\right|^{p}d\mu\\  & \ge\sum_{n\in\mathbb{Z}}\left(2^{np}\right)\mu\left(A_{n}\right)\\  & =\sum_{n\in\mathbb{Z}}2^{np}\left(\lambda_{f}\left(2^{n}\right)-\lambda_{f}\left(2^{n+1}\right)\right). \end{align*}

So

    \[ \sum_{n\in\mathbb{Z}}2^{np}\lambda_{f}\left(2^{n}\right)\le\norm fp^{p}+\sum_{n\in\mathbb{Z}}\lambda_{f}\left(2^{n+1}\right)2^{np}. \]

Hence,

    \[ \sum_{n\in\mathbb{Z}}\left(1-\frac{1}{2^{p}}\right)\lambda_{f}\left(2^{n}\right)2^{np}\le\norm fp^{p}. \]

Take c_{p}=1-\frac{1}{2^{p}}>0. Then we finally get \norm fp^{p}\sim_{p}\sum_{n\in\mathbb{Z}}\lambda_{f}\left(2^{n}\right)2^{np}.

Some interpolation theorems

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/host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1527 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1528 Warning: fclose() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1529 Warning: fopen(/host/home3/willkwon/html/wp-content/ql-cache/quicklatex.com-572690eca68f9c8e7cc0e6a9e9f232c9_l3.png): failed to open stream: Disk quota exceeded in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1527 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1528 Warning: fclose() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1529 Warning: fopen(/host/home3/willkwon/html/wp-content/ql-cache/quicklatex.com-1d4afa78451c8c6248ec176d6e03ae87_l3.txt): failed to open stream: Disk quota exceeded in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1513 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1514 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1515 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1516 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1517 Warning: fclose() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1518 Warning: fopen(/host/home3/willkwon/html/wp-content/ql-cache/quicklatex.com-444da0ebd11234165eaf759e6b4b5506_l3.txt): failed to open stream: Disk quota exceeded in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1513 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1514 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1515 Warning: fwrite() expects parameter 1 to be resource, boolean given in /host/home3/willkwon/html/wp-content/plugins/wp-quicklatex/wp-quicklatex.php on line 1516