\documentclass[11pt]{article}
\usepackage[russian,greek,english]{babel}
\usepackage{amsmath,ifthen,color,eurosym}
\usepackage{wrapfig,hyperref,cite}
\usepackage{latexsym,amsthm,amsmath,amssymb,url,cite}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{stmaryrd,wasysym}
\usepackage{colortbl}
\usepackage[lined,boxed,commentsnumbered]{algorithm2e}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}

\def\baselinestretch{1}


\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newtheorem{claim}{Claim}
\newtheorem{observation}{Observation}
\newtheorem{proposition}{Proposition}

\setlength{\textheight}{22cm}
\setlength{\topmargin}{-1cm}
\setlength{\textwidth}{165mm}
\setlength{\oddsidemargin}{-3mm}
\setlength{\unitlength}{1mm}


%\renewcommand{\refname}{\small References}
\addto{\captionsenglish}{\renewcommand{\refname}{\normalsize
    References}}



%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
%Put here the package you need, new commands, ...



%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%

\begin{document}
%1 page max !!

%Write your title here:
\noindent{\bf\Large $1$-Sperner hypergraphs and new characterizations of threshold graphs}

%List the authors as follows (you may underline the speaker, or just
%name the speaker, and mention your co-authors at the end, up to you):
\noindent E.~Boros -- {\small Rutgers University}

\noindent V.~Gurvich -- {\small Rutgers University}

\noindent \underline{M.~Milani\v c} -- {\small University of Primorska}



\vspace{2ex}

\noindent{\bf Abstract:} %Here write your abstract

We introduce a new class of hypergraphs, the class of $1$-Sperner hypergraphs.
A hypergraph ${\cal H}$ is said to be {\it $1$-Sperner} if every two distinct hyperedges $e,f$ of ${\cal H}$
\hbox{satisfy} $\min\{|e\setminus f|,|f\setminus e|\} = 1$.
We prove a decomposition theorem for $1$-Sperner hypergraphs and examine several of its consequences, including
bounds on the size of $1$-Sperner hypergraphs and a new, constructive proof of the fact that every $1$-Sperner
hypergraph is threshold. (A hypergraph ${\cal H}=(V, {\cal E}) $ is said to be {\it threshold} if there exist a non-negative integer weight function $w:V\to \mathbb{Z}_+$ and a non-negative threshold $t\in \mathbb{Z}_+$ such that for every subset $X\subseteq V$, we have
$\sum_{x\in X}w(x)\ge t$ if and only if $X$ contains a hyperedge.)

A hypergraph is said to be {\it Sperner} (or: {\it a clutter}) if no hyperedge is contained in another one.
Given a graph $G$, the {\it clique hypergraph of $G$} is the hypergraph ${\cal C}(G)$ with vertex set $V(G)$
in which the hyperedges are exactly the maximal cliques of $G$. The clique hypergraphs of graphs are known to be
exactly those Sperner hypergraphs ${\cal H}$ that are also {\it normal} (or: {\it conformal}), that is, for every set $X\subseteq V({\cal H})$ such that every pair of elements in $X$ is contained in a hyperedge, there exists a hyperedge containing $X$.
We show that within the class of normal Sperner hypergraphs, the (generally properly nested) classes of $1$-Sperner hypergraphs, of threshold hypergraphs, and of $2$-asummable hypergraphs coincide. This yields new characterizations of the class of threshold graphs.

The talk is based on~\cite{BGM15}.\footnote{Remark: The manuscript should be soon (hopefully before GROW) posted on arXiv, in which case the reference can be updated accordingly.}

\small

\bigskip

\begin{thebibliography}{1}

\bibitem{BGM15}
E.~Boros, V.~Gurvich, and M.~Milani\v c.
\newblock $1$-Sperner hypergraphs.
\newblock Manuscript, 2015.
\end{thebibliography}
\end{document} 