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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.
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.
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.
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 $S^{-1}S.$
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, ....).
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.
Comment: 8 pages
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