Author:
BRANDT AXEL,IRWIN DAVID,JIANG TAO
Abstract
Given a family ofr-uniform hypergraphs${\cal F}$(orr-graphs for brevity), the Turán number ex(n,${\cal F})$of${\cal F}$is the maximum number of edges in anr-graph onnvertices that does not contain any member of${\cal F}$. A pair {u,v} iscoveredin a hypergraphGif some edge ofGcontains {u, v}. Given anr-graphFand a positive integerp⩾n(F), wheren(F) denotes the number of vertices inF, letHFpdenote ther-graph obtained as follows. Label the vertices ofFasv1,. . .,vn(F). Add new verticesvn(F)+1,. . .,vp. For each pair of verticesvi, vjnot covered inF, add a setBi,jofr− 2 new vertices and the edge {vi, vj} ∪Bi,j, where theBi,jare pairwise disjoint over all such pairs {i, j}. We callHFpthe expanded p-clique with an embedded F. For a relatively large family ofF, we show that for all sufficiently largen, ex(n,HFp) = |Tr(n, p− 1)|, whereTr(n, p− 1) is the balanced complete (p− 1)-partiter-graph onnvertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Turán number is exactly determined (for largen).
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science
Cited by
24 articles.
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