Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a family F of r-graphs, let ex(n, F) be the maximum number of edges in an n-vertex r-graph containing no member of F. Let C(r) 4 denote the family of r-graphs with distinct edges A, B, C, D, such that A ∩ B = C ∩ D = O and A ∪ B = C ∪ D. For s1 l … l sr, let K(r) (s1,…,sr) be the complete r-partite r-graph with parts of sizes s1,…,sr.Furedi conjectured over 15 years ago that ex(n,C(3)4) l (n2) for sufficiently large n.  prove the aker resultex(n, {C(3)4, K(3)(1,2,4)}) l (n2).Generalizing a ll-known conjecture for the Turan number of bipartite graphs,  conjecture thatex(n, K(r)(s1,…,sr)) = Θ(nr−1/s),where s = Πr−1i=1 si.  prove this conjecture when s1 = … = sr−2 = 1 and(i) sr−1 = 2, (ii) sr−1 = sr = 3, (iii)sr > (sr−1−1)!.In cases (i) and (ii),  determine the asymptotic value of ex(n,K(r)(s1,…,sr)). also provide an explicit construction givingex(n,K(3)(2,2,3)) > (1/6−o(1))n8/3.This improves upon the previous best lor bound of O(n29/11) obtained by probabilistic methods. Several related open problems are also presented.
['Dhruv Mubayi']
I am writing a research paper on a idea Some Exact Results and New Asymptotics for Hypergraph Turán Numbers
[{"from": "human", "value": "I am writing a research paper on a idea Some Exact Results and New Asymptotics for Hypergraph Tur\u00e1n Numbers"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a family F of r-graphs, let ex(n, F) be the maximum number of edges in an n-vertex r-graph containing no member of F. Let C(r) 4 denote the family of r-graphs with distinct edges A, B, C, D, such that A \u2229 B = C \u2229 D = O and A \u222a B = C \u222a D. For s1 l \u2026 l sr, let K(r) (s1,\u2026,sr) be the complete r-partite r-graph with parts of sizes s1,\u2026,sr.Furedi conjectured over 15 years ago that ex(n,C(3)4) l (n2) for sufficiently large n.  prove the aker resultex(n, {C(3)4, K(3)(1,2,4)}) l (n2).Generalizing a ll-known conjecture for the Turan number of bipartite graphs,  conjecture thatex(n, K(r)(s1,\u2026,sr)) = \u0398(nr\u22121/s),where s = \u03a0r\u22121i=1 si.  prove this conjecture when s1 = \u2026 = sr\u22122 = 1 and(i) sr\u22121 = 2, (ii) sr\u22121 = sr = 3, (iii)sr > (sr\u22121\u22121)!.In cases (i) and (ii),  determine the asymptotic value of ex(n,K(r)(s1,\u2026,sr)). also provide an explicit construction givingex(n,K(3)(2,2,3)) > (1/6\u2212o(1))n8/3.This improves upon the previous best lor bound of O(n29/11) obtained by probabilistic methods. Several related open problems are also presented."}]