Behrend, F. A. On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. U. S. A. 32 (1946), 331-332.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....proper k colorability, currently there is no known test (with a constant number of queries) for the property of being triangle free that has a better dependency on . In fact, it is known that the number of queries of a 1 sided test for this property cannot be polynomial in , using a bound from [12]. When one tries to generalize the above result to properties de ned as not containing a xed given induced subgraph, other than a clique, another problem arises (an induced subgraph is a subgraph obtained from the original graph by deleting vertices and their incident edges, but deleting no ....

....of course independent of the input size) is given using a graph that is far from being triangle free, but still does not contain too many distinct triangles, so q queries will not capture a triangle with high probability. Such a graph can be constructed using the number theoretic construction in [12]; the details and a generalization thereof to other properties de ned in terms of not containing a xed (not necessarily induced) subgraph are found in [1] As the main example here for using Yao s method (and one that concerns 2 sided algorithms as well) we consider the following problem: ....

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proceedings of the National Academy of Sciences of the United States of America 32 (1946), 331-332. 27

....few copies of H. The proof of this part, described in Section 6 uses the approach of [1] but requires some additional ideas. It applies some properties of digraph homomorphisms as well as certain constructions in additive number theory, based on (simple variants of) the construction of Behrend [11] of dense subsets of the first n integers without three term arithmetic progressions. In Section 7 we describe the proof of Theorem 4. We assume, throughout these three sections, that the underlying undirected graph of the digraph H considered is connected. In the final section, Section 8, we ....

....we obtain a lower bound of #(1 #) as required. 6 Hard to Test Digraphs In this section we apply the approach used in [1] together with some additional ideas, in order to prove Theorem 2 part (iii) This approach uses techniques from additive number theory, based on the construction of Behrend [11] of dense sets of integers with no three term arithmetic progressions, together with some properties of homomorphisms of digraphs. A linear equation with integer coe#cients a i x i = 0 (5) in the unknowns x i is homogeneous if a i = 0. If X 2, m , we say that X has no ....

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. National Academy of Sciences USA 32 (1946), 331--332.

....the full version of the Regularity Lemma is required for the proofs, as follows from the main result of [9] For properties shown testable in [8] testers whose number of queries is polynomial in 1 are given. However, it is worth noting that using the number theoretic construction of Behrend [4] we can show that any one sided error test for some simple properties, like being triangle free, requires a number of queries which cannot be bounded by a polynomial in 1 (by the existence of graphs which are far from being triangle free and yet do not contain too many distinct triangles) ....

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. National Academy of Sciences USA 32 (1946), 331-332.

....the full version of the Regularity Lemma is required for the proofs, as follows from the main result of [9] For properties shown testable in [8] testers whose number of queries is polynomial in 1 are given. However, it is worth noting that using the number theoretic construction of Behrend [4] we can show that any one sided error test for some simple properties, like being triangle free, requires a number of queries which cannot be bounded by a polynomial in 1 (by the existence of graphs which are far from being triangle free and yet do not contain too many distinct ....

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. National Academy of Sciences USA 32 (1946), 331--332.

....proof and the known bounds in the proof of the regularity lemma actually show that f(n) O n 2 (log n) 1=5 ; where log n = minfk j log 2 log 2 : log 2 z k times n 1g. 3 The lower bound Our main tool here is an arithmetic lemma proven using the method of Behrend [2], and its extension by Ruzsa [4] with some modi cations. A linear equation with integer coecients X a i x i = 0 (4) in the unknowns x i is homogeneous if P a i = 0. If X N = f1; 2; ng, we say that X has no non trivial solution to (4) if whenever x i 2 X and P a i x i = 0, it ....

....5)w = 0; 5) 2. a set X 2 N , jX 2 j n 2 O(log 3=4 n) with no non trivial solution to 5x (q 3)y 3z (q 5)w = 0; 6) 3. a set X 3 N , jX 3 j n 2 O(log 3=4 n) with no non trivial solution to 5x qy 2z (q 3)w = 0: 7) Proof. To prove part 1 we apply the method of Behrend [2]. Let d be an integer (to be chosen later) and de ne X 1 = f k X i=0 x i d i j x i d q 5 (0 i k) k X i=0 x 2 i = Bg; where k = blog n= log dc 1 and B is chosen to maximize the cardinality of X 1 . If x; y; z; w 2 X 1 satisfy (5) and x = k X i=0 x i d i ; y = k ....

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. National Academy of Sciences USA 32 (1946), 331-332.

....of the Regularity Lemma is required for the proofs, as follows from the main result of [9] For properties shown testable in [8] ffl testers whose number of queries is polynomial in ffl Gamma1 are given. However, it is worth noting that using the number theoretic construction of Behrend [4] we can show that any one sided error ffl test for some simple properties, like being triangle free, requires 22 a number of queries which cannot be bounded by a polynomial in ffl Gamma1 (by the existence of graphs which are ffl far from being triangle free and yet do not contain too many ....

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. National Academy of Sciences USA 32 (1946), 331--332.

No context found.

Behrend, F. A. On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. U. S. A. 32 (1946), 331-332.

No context found.

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proceedings of the National Academy of Sciences of the United States of America 32 (1946), 331--332.

No context found.

F. A. Behrend. On sets of integers which contain no three terms in arithmetic progression. In Proceedings of the National Academy of Sciences of the United States of America 32, 331--332, 1946.

No context found.

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. National Academy of Sciences USA 32 (1946), 331--332.

No context found.

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. National Academy of Sciences USA 32 (1946), 331--332.

No context found.

F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. National Academy of Sciences USA 32:331-332, 1946.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC