ヒトES細胞からの眼杯および保存可能な多層網膜組織の自己組織化

京都大学0048新制・論文博士博士(医学)乙第12800号論医博第2072号新制||医||1001(附属図書館)80844(主査)教授 髙橋 淳, 教授 吉村 長久, 教授 江藤 浩之学位規則第4条第2項該当Doctor of Medical ScienceKyoto UniversityDFA

Two-colorings with many monochromatic cliques in both colors

Color the edges of the n-vertex complete graph in red and blue, and suppose that red k-cliques are fewer than blue k-cliques. We show that the number of red k-cliques is always less than cknk, where ck∈(0, 1) is the unique root of the equation zk=(1-z)k+kz(1-z)k-1. On the other hand, we construct a coloring in which there are at least cknk-O(nk-1) red k-cliques and at least the same number of blue k-cliques. © 2013 Elsevier Inc

Non-trivial 3-wise intersecting uniform families

A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results shows that for every $\epsilon>0$ there exists $n_0$ such that if $n>n_0$ and $\frac25+\epsilon<\frac kn<\frac 12-\epsilon$ then the maximum size is $4\binom{n-4}{k-3}+\binom{n-4}{k-4}$.Comment: 12 page

Strong stability of 3-wise tt-intersecting families

Let $\mathcal G$ be a family of subsets of an $n$-element set. The family $\mathcal G$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in $\mathcal G$ is of size at least $t$. For a real number $p\in(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\in\mathcal G$. For example, if $\mathcal G$ consists of all subsets containing a fixed $t$-element set, then it is a $3$-wise $t$-intersecting family with the measure $p^t$. For a given $\delta>0$, by choosing $t$ sufficiently large, the following holds for all $p$ with $0<p\leq 2/(\sqrt{4t+9}-1)$. If $\mathcal G$ is a $3$-wise $t$-intersecting family with the measure at least $(\frac12+\delta)p^t$, then $\mathcal G$ satisfies one of (i) and (ii): (i) every subset in $\mathcal G$ contains a fixed $t$-element set, (ii) every subset in $\mathcal G$ contains at least $t+2$ elements from a fixed $(t+3)$-element set