Bounds of finite relations

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}$$\end{document}be a finite set of relations of degree n. A number N is found such that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}$$\end{document}has cardinality less than N then for every relation R of degree n the following holds: if no member of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}$$\end{document}is embeddable in R then R has a strict extension in which no member of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}$$\end{document}is embeddable. In fact, if all members of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}$$\end{document}are defined on sets with at least n elements this number N is 3s(n,2). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {II}\limits_{k = 3}^n$$\end{document} 2s(n,k).k-1 where s(n,k) are the Stirling numbers of the second kind.