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15 records were found.

## On iterated image size for point-symmetric relations

Let $\Gamma =(V,E)$ be a point-symmetric reflexive relation and let $v\in V$ such that $|\Gamma (v)|$ is finite (and hence $|\Gamma (x)|$ is finite for all $x$, by the transitive action of the group of automorphisms). Let $j\in \N$ be an integer such that $\Gamma ^j(v)\cap \Gamma ^{-}(v)=\{v\}$. Our main result states that $$|\Gamma ^{j} (v)|\ge | \Gamma ^{j-1} (v)| + |\Gamma (v)|-1.$$ As an application we have $|\Gamma ^{j} (v)| \ge 1+(|\Gamma (v)|-1)j.$ The last result confirms a recent conjecture of Seymour in the case of vertex-symmetric graphs. Also it gives a short proof for the validity of the Caccetta-H\"aggkvist conjecture for vertex-symmetric graphs and generalizes an additive result of Shepherdson.

## Some additive applications of the isopermetric approach

Comment: 28 pages

## A Structure Theory for Small Sum Subsets

Comment: 29 pages

## On Group bijections $\phi$ with $\phi(B)=A$ and $\forall a\in B, a\phi(a) \notin A$

A {\em Wakeford pairing} from $S$ onto $T$ is a bijection $\phi : S \to T$ such that $x\phi(x)\notin T,$ for every $x\in S.$ The number of such pairings will be denoted by $\mu(S,T)$. Let $A$ and $B$ be finite subsets of a group $G$ with $1\notin B$ and $|A|=|B|.$ Also assume that the order of every element of $B$ is $\ge |B|$. Extending results due to Losonczy and Eliahou-Lecouvey, we show that $\mu(B,A)\neq 0.$ Moreover we show that $\mu(B,A)\ge \min \{\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}\},$ unless there is $a\in A$ such that $|Aa^{-1}\cap B|=|B|-1$ or $Aa^{-1}$ is a progression. In particular, either $\mu(B,B) \ge \min \{\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}\},$ or for some $a\in B,$ $Ba^{-1}$ is a progression.

## Extensions of the Moser-Scherck-Kemperman-Wehn Theorem

Let $\Gamma =(V,E)$ be a reflexive relation having a transitive group of automorphisms and let $v\in V.$ Let $F$ be a subset of $V$ with $F\cap \Gamma ^-(v)=\{v\}$. (i) If $F$ is finite, then $| \Gamma (F)\setminus F|\ge |\Gamma (v)|-1.$ (ii) If $F$ is cofinite, then $| \Gamma (F)\setminus F|\ge |\Gamma ^- (v)|-1.$ In particular, let $G$ be group, $B$ be a finite subset of $G$ and let $F$ be a finite or a cofinite subset of $G$ such that $F\cap B^{-1}=\{1\}$. Then $| (FB)\setminus F|\ge |B|-1.$ The last result (for $F$ finite), is famous Moser-Scherck-Kemperman-Wehn Theorem. Its extension to cofinite subsets seems new. We give also few applications.

## On Minkowski product size: The Vosper's property

A subset $S$ of a group $G$ is said to be a Vosper's subset if $|A\cup AS|\ge \min (|G|-1,|A|+|S|),$ for any subset $A$ of $G$ with $|A|\ge 2.$ In the present work, we describe Vosper's subsets. Assuming that $S$ is not a progression and that $|S^{-1} S|, |S S^{-1}| <2 |S|,|G'|-1,$ we show that there exist an element $a\in S,$ and a non-null subgroup $H$ of $G'$ such that either $S^{-1}HS =S^{-1}S \cup a^{-1}Ha$ or $SHS^{-1} =SS^{-1}\cup aHa^{-1},$ where $G'$ is the subgroup generated by $S^{-1}S.$

## Topology of Cayley Graphs Applied to Inverse Additive Problems

We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where $1\le r \le k.$ These results may be applied to several questions in Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem in finite fields and Cayley graphs with a big diameter, ....).

## On the size of spheres of relations with a transitive group of automorphisms

Let $\Gamma =(V,E)$ be a point-transitive reflexive relation. Let $v\in V$ and put $r=|\Gamma (v)|.$ Also assume $\Gamma ^j(v)\cap \Gamma ^{-}(v)=\{v\}$. Then $$|\Gamma ^{j} (v)\setminus \Gamma ^{j-1} (v)| \ge r-1.$$ In particular we have $|\Gamma ^{j} (v)| \ge 1+(r-1)j.$ The last result confirms a recent conjecture of Seymour in the case vertex-transitive graphs. Also it gives a short proof for the validity of the Caccetta-H\"aggkvist conjecture for vertex-transitive graphs and generalizes an additive result of Shepherdson.

## Hyper-atoms and the critical pair Theory

Comment: 16 pages

Comment: 8 pages