Name
Abstract | ||
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The Turdn number of an r-uniform graph F, denoted by ex(n, F), is the maximum number of edges in an F-free r-uniform graph on n vertices. The Turdn density of F is defined as pi(F) = lim(n ->infinity) ex(n, F)/(rn). Denote Pi((r))(infinity) = {pi(F) : F is a family of r-uniform graphs}, Pi((r))(fin) = {pi(F) : F is a finite family of r-uniform graphs}, and Pi((r))(infinity) = {pi(F) : F is a family of r-uniform graphs, and vertical bar F vertical bar <= t}. For graphs, Erdos and Simonovits [Stadia Sci. Mat. Hungar. 1 (1966), pp. 51-57] and Erdos and Stone [Bull. Amer. Math. Soc., 52 (1946), pp. 1087-1091] showed that Pi((2))(infinity) = Pi((2))(fin) = Pi((2))(1) = {0, 1/2, 2/3, ..., l-1/l, ...}. We know quite little about the Turan density of an fin r-uniform graph for r >= 3. Baber and Talbot [Electron. J. Combin., 19 (2011)] and Pikhurko [Israel J. Math., 20 (2014), pp. 415-454] showed that there is an irrational number in Pi((3))(3) and Pi((3))(fin), respectively, disproving a conjecture of Chung and Graham [Erdos on Graphs: His Legacy of Unsolved Problems, A. K. Peters, Natick, MA, 1999]. Baber and Talbot [Electron. J. Combin., 19 (2011)] asked whether Pi((r))(1) contains an irrational number. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. The Lagrangian density of an r-uniform graph F is pi(lambda)(F) = sup{r!lambda(G) : G is F-free}, where lambda(G) is the Lagrangian of an r-uniform graph G. Sidorenko [Combinatorica, 9 (1989), pp. 207-215] showed that the Lagrangian density of an r-uniform hypergraph F is the same as the Turan density of the extension of F. In this paper, we show that the Lagrangian density of F = {123, 124, 134, 234, 567} (the disjoint union of K-4(3) and an edge) is root 3/3, and consequently, the Turan density of the extension of F is an irrational number, answering the question of Baber and Talbot. |
Year | DOI | Venue |
|---|---|---|
2022 | 10.1137/21M1410798 | SIAM JOURNAL ON DISCRETE MATHEMATICS |
Keywords | DocType | Volume |
hypergraph Lagrangian, Lagrangian density, Turan density | Journal | 36 |
Issue | ISSN | Citations |
1 | 0895-4801 | 0 |
PageRank | References | Authors |
0.34 | 0 | 2 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Zilong Yan | 1 | 0 | 0.34 |
| Yuejian Peng | 2 | 2 | 3.46 |