A-Systems, Independent Functions, and Sets Bounded in Spaces by Novikov S.Y.

By Novikov S.Y.

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For such a graph, how many vertices (edges) must lie on triangles and, more generally, on cycles of length 2k + 1? 5 ([31)). Let G be a graph with n vertices and at least Ln 2 /4J + 1 edges. (a) At least Ln/2J +2 vertices of G are on triangles, and this result is sharp. (b) At least 2Ln/2J + 1 edges of G are on triangles, and this result is sharp. (c) If k 2 2 and n 2 max{3k(3k + 1), 216(3k - 2)}, then at least 2(n - k)/3 vertices of G are on cycles of length 2k + 1, and this result is asymptotically best possible.

E. Conf. , Graph Theory and Computing 283-294, (1982). 36. P. Erdos, R. J. Faudree, C. C. Rousseau, and R. H. Schelp, Tree - multipartite graph Ramsey numbers, Graph Theory and Combinatorics - A Volume in Honor of Paul Erdos, Bela Bollobas, editor, Academic Press, (1984), 155-160. 26 R. J. Faudree, C. C. Rousseau and R. H. Schelp 37. P. Erdos, R. J. Faudree, C. C. Rousseau, and R. H. Schelp, Multipartite graph - sparse graph Ramsey numbers, Combinatorica 5, (1985), 311-318. (with P. Erdos, C. C.

If F--+ (nG), then must F contain at least liJ7g1,J copies of G? We have shown that if G and E > 0 are fixed then for all sufficiently large n, every graph F satisfying F --+ (nG) contains at least r(nG)(I- E)IW(G)I copies of G. 2 Ramsey Minimal Problems Let F, G, H be graphs without isolates. The graph F is (G, H)-minimal if F --+ (G,H) but F - e f> (G,H) for any edge e of F. The pair (G,H) is called Ramsey-finite or Ramsey-infinite according to whether the class of all (G, H)minimal graphs is a finite or infinite set.

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