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Subgraph densities in hypergraphs

100%

Peng Y.

Discussiones Mathematicae Graph Theory

2007

tom 27

nr 2

281-297

Let r ≥ 2 be an integer. A real number α ∈ [0,1) is a jump for r if for any ε > 0 and any integer m ≥ r, any r-uniform graph with n > n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c = c(α) > 0 does not depend on ε and m. A result of Erdös, Stone and Simonovits implies that every α ∈ [0,1) is a jump for r = 2. Erdös asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumps for every r ≥ 3. However, there are still a lot of open questions on determining whether or not a number is a jump for r ≥ 3. In this paper, we first find an infinite sequence of non-jumps for r = 4, then extend one of them to every r ≥ 4. Our approach is based on the techniques developed by Frankl and Rödl.

Lagrangians and Euler morphisms on fibered-fibered frame bundles from projectable-projectable classical linear connections

88%

Bednarska A.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2011

tom 65

nr 1

We classify all \(\mathcal{F}^2\mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operators \(A\) transforming projectable-projectable torsion-free classical linear connections \(\nabla\) on fibered-fibered manifolds \(Y\) of dimension \((m_1,m_2, n_1, n_2)\) into \(r\)th order Lagrangians \(A(r)\) on the fibered-fibered linear frame bundle \(L^{fib-fib}(Y )\) on \(Y\). Moreover, we classify all \(\mathcal{F}^2\mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operators \(B\) transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds \(Y\) of dimension \((m_1,m_2, n_1, n_2)\) into Euler morphism \(B(\nabla)\) on \(L^{fib-fib}(Y )\). These classifications can be expanded on the \(k\)th order fibered-fibered frame bundle \(L^{fib-fib,k}(Y )\) instead of \(L^{fib-fib}(Y )\).

Ograniczanie wyników

1 Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

1 Discussiones Mathematicae Graph Theory

1 Bednarska A.

1 Peng Y.

1 2011

1 2007

Wyniki wyszukiwania

Subgraph densities in hypergraphs

Lagrangians and Euler morphisms on fibered-fibered frame bundles from projectable-projectable classical linear connections