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• W3149159487 abstract "1 . I NTRODUCTION In combinatorics we often meet two kinds of extremal problems . In one kind , optimal configurations consist of 'objects' , which are somehow uniformly spread in the space under consideration ; and in the other kind , optimal configurations consist of 'objects' , which are somehow compressed . To the first kind belong packing , covering and coding problems , whereas diametric (especially of Erdos - Ko - Rado type) , vertex- and edge-isoperimetric problems belong to the second kind . For many problems of the spreading type , the probabilistic method gives good or even asymptotically optimal results but , mostly , strictly optimal configurations are unknown . In contrast , problems of the compressing type can often be solved exactly with pushing techniques ('pushing down' , 'pushing to the left' etc . ; see (14)) . However , the success of pushing operations is linked to the property that there is a 'nested' structure of optimal configurations with respect to some order . When this is not the case , then there are competing configurations (for example , in (6)) and solutions are harder to obtain . We concentrate here on edge-isoperimetric problems . They can be defined for any graph G 5 ( 9 , % ) as follows . For any A ' 9 , define the set @ ( A ) of all boundary edges ; that is , @ ( A ) 5 hh x , y j P % : u h x , y j > A u 5 1 j ." @default.
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• W3149159487 title "General Edge-isoperimetric Inequalities , Part I : Information-theoretical Methods" @default.
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