Báo cáo toán học: "A characterization for sparse ε-regular pairs" pps

... in a similar way as described in [1] for ε-regularity, that one can find in time n O(1/p) a partition guaranteed by the version of Szemer´edi’s lemma for sparse graphs that was introduced by Kohayakawa ... Theorem 2.2 for the precise statement. 1.1 Notation For a graph G = (V, E) and sets A, B ⊂ V , we write e G (A, B) for the number of edges with one endpoint in A and one endpoint...

Ngày tải lên: 07/08/2014, 15:22

... π  2 − 1, α i = π i+1 − 1 for 1 ≤ i ≤ m and α  i = π  i+1 − 1for1 ≤ i ≤ d(π) − 1. Clearly, f is a bijection between P  (n, k, l)and P(n −k − l +1,k−1,l−1) Furthermore, for 1 ≤ j ≤ d(π), s + j  i=1 (π  i −π i )=(s ... definition, for t  ≥ t ≥ 0, P(n, k, l, t) ⊆ P (n, k, l, t  ), thus for any s ≥ n, P (n, k, l, n) ⊆ P (n, k, l, s). The result follows since P(n, k, l, s) ⊆ P(n, k,...

Ngày tải lên: 07/08/2014, 06:20

... M r (g)=(M(|g| r )) 1/r . For (25), we choose r so that n/a < r < β. By Lemma 4, we have (14). For (26), we choose r so that n/a < r < min(q, β). By Lemma 4, we have (15). A Characterization ... for some N 0 > 0 d j = O(2 jN 0 ),j→∞, and that for any N>0, and j ∈ N 0 = N ∪{0}, there exists a constant C N independent of j such that d j  C N ∞  k=j 2 (j−k)N b k d 1...

Ngày tải lên: 06/08/2014, 05:20

... interesting to note that the expansions for β(A) in terms of convex subsets generally have a simpler form than other expansions. In particular, the forms given for β(A) in Propositions 4.1, 4.2, 4.5, ... B k .Then β(A)=  k>0 (−2) k−1 kf k . The next result gives an expansion for β(A) for an antimatroid A which is similar to the M¨obius function formulation for a matroid. (See Se...

Ngày tải lên: 07/08/2014, 06:20

... the unimodality of the bino- mial coefficients. Example 2 .For0 ≤ k ≤ m − 1, define b k (m):=2 −2m+k  2m − 2k m − k  m + k m  (a +1) k for 0 ≤ k ≤ m − 1. Then b k+1 (m) b k (m) = (m − k)(m + k ... encountered P m in the integral formula [1]  ∞ 0 dx (x 4 +2ax 2 +1) m+1 = πP m (a) 2 m+3/2 (a +1) m+1/2 . (3.1) This does not appear in the standard tables. Example 3 .For0 ≤ k ≤ m − 1, define b...

Ngày tải lên: 07/08/2014, 06:20

... o(1). 2.1 Sparse Non Cover Graphs: proof of Theorem 1.4 For positive integers a, b, a partition V 0 ,V 1 , ,V a of an n-set is called an (a, b)- partition if |V 0 | = b and |V i | = |V j | for all ... recently proved by R¨odl and Thoma [10]. For sparse random graphs, the authors of [4] show that whp c(G n,p )=o(e(G n,p )). Namely they prove the following two bounds. For every c ≥...

Ngày tải lên: 07/08/2014, 06:20

... + j for 1 ≤ j ≤ n 2(4 j−1 )ifi=n−jfor 1 ≤ j ≤ n −1. For i greater than 2n (the second region of definition), d n (i) is defined recursively; in particular, d n (i)= i−1  j=i−b n (i) d n (j) for ... electronic journal of combinatorics 5(1998), #R3 13 as (8) implies e(k) ≥ 0 for k ≥ 0. It is easy to see that (8) holds for k =0,1,2. For k ≥ 3 we proceed by induction: e(k +1)=e(k)− e(k...

Ngày tải lên: 07/08/2014, 06:22

... follows from the fact that c(m, n) → c as n → ∞. References [1] H. Bielak, Size Ramsey numbers for some regular graphs, Electronic Notes in Discrete Math. 24 (2006), 39–45. [2] P. Erd˝os, R. ... number of a complete bipartite graph, Discrete Math. 113 (1993), 259–262. [4] I. Gorgol, On bounds for size Ramsey numbers of a complete tripartite graph, Discrete Math. 164 (1997), 149–153. the ....

Ngày tải lên: 07/08/2014, 13:21

... Arnoux-Rauzy sequence which is not c-balanced for any c. A sequence s ∈ A ∞ is c-balanced if for all factors u and v of s having the same length, one has ||u| a − |v| a | ≤ c for every a ∈ A Corollary 3.14. ... (n + 1) factors of length n, for every n. For Sturmian sequences, the two conditions are equivalent. There are two different generalizations of Sturmian sequences for alphab...

Ngày tải lên: 07/08/2014, 15:22

... al. [7] used this recursion to derive a simple recursion for t g which leads to an asymptotic formula for t g . This asymptotic formula for t g was used in [ 20] to settle a conjecture of  t ... c(r)[d(r)] g p g . (8) We note that the above formulas easily lead to asymptotic formulas for t g (r) and p g (r) (as g → ∞), using the corresponding asymptotic formulas for t g and p g . F...

Ngày tải lên: 08/08/2014, 12:23