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More Sets, Graphs and Numbers: A Salute to Vera Sòs and - download pdf or read online

By Ervin Gyori, Gyula O.H. Katona, László Lovász

ISBN-10: 3540323775

ISBN-13: 9783540323778

ISBN-10: 9639453056

ISBN-13: 9789639453050

This quantity honours the eminent mathematicians Vera Sos and Andras Hajnal. The ebook comprises survey articles reviewing classical theorems, in addition to new, cutting-edge effects. additionally provided are leading edge expository learn papers with new theorems and proofs within the region of the classical Hungarian topics, like extremal combinatorics, colors, combinatorial quantity idea, and so forth. The open difficulties and the most recent leads to the papers are bound to encourage extra research.

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Additional info for More Sets, Graphs and Numbers: A Salute to Vera Sòs and András Hajnal

Sample text

Billl and D. ) In t he sequel, constants implied by "0 (·)", "«" and "» " may depend only on F , G and E . 7:, Y( x)) . By Eisenstein's t heorem, for every v E MQ th ere exists Cv 2: 1 such that all but finitely many cv are equal to 1, and the j j coefficients of t he power series Y( x) = 2:~I OjX and U(x) = 2:~nN bjx satisfy (12) For 0: E Q put V'(o:) = Since - { v E V(o:) : 100I v 2:vEVII(Q) log 100Iv < 1/(2cv ) if v = 00, } < 1/Cv if v < 00 . - « - L:vEV II(Q) log 100I v logH(o:) Hence it is sufficient to find a partition V' (0:) log 100I v , log H (0:) - L:vEV' V"(o:) = V(o:) \ V'(o:) .

For a partition V(G) f(X, Y) = = Xu Y, let L us write Id~Ll(K, Y) - pd~~L(K, Y)I · I

This immedi at ely impli es Theorem 6. Ind eed , for non-ar chimedean v one may pu t Cv = ITI;;-l . For archimedean v , t he existe nce of Cv follows from th e fact t ha t th e convergence radius of a complex algebraic power series is positive. Eisenstein 's th eorem goes back to Eisenstein 's pap er [8]. See [10, page 151] for an old-fashioned pro of and [7] for a mod ern quantit ative argument. See also [2, page 28] for an especially quick proof when J( = Q, which suffices for t he present note.

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More Sets, Graphs and Numbers: A Salute to Vera Sòs and András Hajnal by Ervin Gyori, Gyula O.H. Katona, László Lovász


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