We consider a variety of problems in extremal graph and set theory.Given a property $\Gamma$ and a family of sets${\mathcal F}$, let $f({\mathcal F},\Gamma)$ be the size of the largest subfamily of ${\mathcal F}$ having property $\Gamma$. Let $f(m,\Gamma)$ be the minimum of $f({\mathcal F},\Gamma)$ over all families of size $m$ where $m$ is a positive integer. A family $\mathcal{F}$ is {\it $B_d$-free} if it has no subfamily $\mathcal{F}'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinctsets such that for every $I,J \subseteq [d]$, both $F_I \cup F_J=F_{I \cup J}$ and $F_I \cap F_J = F_{I \cap J}$ hold.A family $\mathcal{F}$ is $a$-{\it union-free} if $F_1\cup \dots \cup F_a \neq F_{a+1}$ whenever $F_1,\dots,F_{a+1}$ are distinct sets in $\mathcal{F}$.We prove a conjecture of Erd\H os and Shelah that $f(m, B_2\text{\rm -free})=\Theta(m^{2/3})$.We also obtain lower and upper bounds for $f(m, B_d\text{\rm -free})$ and $f(m,a\text{\rm -union-free})$.A graph $G$ is {\it $F$-saturated } if it does not contain $F$ as a subgraph but the addition of any new edge creates at least one copy of $F$ in $G$. We focus on finding the minimum size of an $n$-vertex$F$-saturated graph, denoted by$\sat(n,F)$. We prove $ \sat(n,C_k) = n + \frac{n}{k}+ O((\frac{n}{k^2}) + k^2)$for all $n\geq k\geq 3$, where $C_k$ is a cycle with length $k$. We conjecture that our three constructions are optimal. We obtain the exact asymptotics for the number of $n$-vertex graphs of diameter$d$, extending earlier results to hold for almost all $d$ and $n$. Additionally, we find the typical structure of almost all $n$-vertex graphs with diameter of at least $d$. In the case $d < n - c_1 \log n$, the typical graph of diameter $d$ consists ofan induced path of length $d$ and a highly connected blockof order $n-d+3$. In the case $d > n - c_2 \log n$, the typical graph has a completely different snake-like structure. We also extend the results to random graphs of diameter $d$ with edge probability $p$.
PDF
download
878987797
028-85220240
