1,023 research outputs found
A proof of the stability of extremal graphs, Simonovits' stability from Szemer\'edi's regularity
The following sharpening of Tur\'an's theorem is proved. Let denote
the complete --partite graph of order having the maximum number of
edges. If is an -vertex -free graph with edges
then there exists an (at most) -chromatic subgraph such that
.
Using this result we present a concise, contemporary proof (i.e., one
applying Szemer\'edi's regularity lemma) for the classical stability result of
Simonovits.Comment: 4 pages plus reference
2-cancellative hypergraphs and codes
A family of sets F (and the corresponding family of 0-1 vectors) is called
t-cancellative if for all distict t+2 members A_1,... A_t and B,C from F the
union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let
c(n,t) be the size of the largest t-cancellative family on n elements, and let
c_k(n,t) denote the largest k-uniform family. We significantly improve the
previous upper bounds, e.g., we show c(n,2) n_0). Using an
algebraic construction we show that the order of magnitude of c_{2k}(n,2) is
n^k for each k (when n goes to infinity).Comment: 20 page
Uniform hypergraphs containing no grids
A hypergraph is called an rĂr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that AiâŠAj=BiâŠBj=Ď for 1â¤i<jâ¤r and {pipe}AiâŠBj{pipe}=1 for 1â¤i, jâ¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1âŠC2{pipe}={pipe}C2âŠC3{pipe}={pipe}C3âŠC1{pipe}=1, C1âŠC2â C1âŠC3. A hypergraph is linear, if {pipe}EâŠF{pipe}â¤1 holds for every pair of edges Eâ F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For râĽ. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Š 2013 Elsevier Ltd
Extremal numbers for odd cycles
We describe the C_{2k+1}-free graphs on n vertices with maximum number of
edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1.
The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall,
and Bollobas, but here we give a new streamlined proof. The complete
determination of the extremal graphs is also new.
We obtain that the bound for n_0(C_{2k+1}) is 4k in the classical theorem of
Simonovits, from which the unique extremal graph is the bipartite Turan graph.Comment: 6 page
A discrete isodiametric result: the Erd\H{o}s-Ko-Rado theorem for multisets
There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new
results (and problems) concerning families of -intersecting -element
multisets of an -set and point out connections to coding theory and
classical geometry. We establish the conjecture that for such
a family can have at most members
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