N. Alon and M. Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs and Combinatorics 13 (1997), 217-225.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....construction [1] provides that = 1) is some kind of a threshold for an (n; d; graph to be 3 Tur an, but it s not clear what happens when t 4. Some construction could be obtained by the slight modi cation of the Erd os R enyi graphs, which appear in the paper of Alon and Krivelevich [2]. These are (n; d; graphs with parameters d = n (1 o(1) and = n 2(t 2) 1 o(1) which can be made K t free by deleting at most n (1 o(1) vertices. It shows that Theorem 1.1 is not true with the weaker condition Cd provided C is a large enough constant. A plausible ....

N. Alon, M. Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs and Combinatorics 13 (1997), 217-225.

....f r;s (n) c 2 n r s 1 (ln n) 1 r 1 ; where c 1 ; c 2 are positive constants depending only on r and s. Most of these results were obtained using probabilistic method. Recently, two nice explicit constructions which provide upper bound on f r;s (n) were obtained by Alon and Krivelevich in [1]. As one can see, the upper bound on f r;s (n) attract a lot of attention and improved considerably during the last forty years. On the other hand not much progress was made on obtaining a good lower bounds. As was pointed out by Bollob as and Hind [2] no essentially nontrivial lower bound was ....

N. Alon and M. Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs and Combinatorics 13 (1997), 217-225.

....of their Laplace matrices. We omit the details. By a special case of Proposition 4. 1, if = O( d) then (G) O(d= ln d) There are many interesting regular graphs with this property (including all three families of examples described in Section 2 above) Other examples appear in, e.g. 17] [3]. Another variant of Proposition 4.1 is obtained by noting that if 1 2 n are the eigenvalues of the adjacency matrix of a d regular graph G containing N cycles of length 4 then i=1 i = 8N nd nd(d 1) c.f. e.g. 7] Chapter 9 for the easy argument) Therefore, if ....

N. Alon and M. Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs and Combinatorics 13 (1997), 217-225.

....only on the values of r and s and in the particular case r = 3; s = 4 f 3;4 (n) cn 2=3 1=3 ; 11) where c is an absolute constant. The bounds (10) and (11) were obtained by combining the local lemma with the Janson inequality( 9] Some constructive bounds for f r;s (n) are presented in [1]. Here we improve the bound (11) using the technique based on large deviation inequalities. Our treatment is rather similar to that of previous section and can be used to improve (10) in a similar manner. 10 Theorem 2 There exists an absolute constant c 0 such that f 3;4 (n) c 0 n Proof. ....

N. Alon and M. Krivelevich, Constructive bounds for a Ramsey-type problem, preprint. 14

....is bounded by 2 q , since for every possible value of x 0 ; x t 1 we have at most two possible choices of x t . Actually using more complicated computation, which we omit, one can determine the exact number of vertices with loops. The eigenvalues of G are easy to compute (see [11]) Indeed, let A be the adjacency matrix of G. Then, by the properties of PG(q; t) A = AA T = J (d q;t )I, where = q , J is the all one matrix and I is the identity matrix, both of size n q;t n q;t . Therefore the largest eigenvalue of A is d q;t and the absolute value of all ....

....the correct number of copies of any xed sparse graph. An additional advantage of this result is that its assertion depends not on the number of vertices s in H but only on its maximum degree which can be smaller than s. Special cases of this result have appeared in various papers including [11], 13] and probably other papers as well. The approach here is similar to the one in [13] Theorem 4.10 [6] Let H be a xed graph with r edges, s vertices and maximum degree , and let G = V; E) be an (n; d; graph, where, say, d 0:9n. Let m n satisfy m ( Then, for every subset ....

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N. Alon and M. Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs and Combinatorics 13 (1997), 217-225.

....for other instances of H, we are unable at this stage to predict the optimal relation between parameters n; d; for most of the choices of H, for example, for the case H = K 4 . This is due to the absence of known optimal constructions of pseudo random graphs without a single copy of H (see [3] for partial results in the case of H being a xed clique) We conjecture however that conditions on n; d; sucient to guarantee a copy of H in a pseudo random graph G, will be already sucient to guarantee the existence of a fractional or even an integer H factor. Finally it is worth mentioning ....

N. Alon and M. Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs and Combinatorics 13 (1997), 217-225.

.... Research supported in part by a grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT 9600919 ME 103 from the NSF (USA) and the M SMT (Czech Republic) 1 Omega Gamma m 2 ) lower bound for R(s; m) for any fixed s and large m, see, e.g. [3] for a construction that supplies a nearly quadratic bound. In the present paper we describe larger explicit lower bounds for R(s; m) for fixed s and large m. Our main result is the following. Theorem 1.1 There exists an 0 and an explicit construction of graphs such that for every s and ....

N. Alon, M. Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs and Combinatorics 13 (1997), 217-225.

....their Laplace matrices. We omit the details. By a special case of Proposition 4. 1, if = O( p d) then (G) O(d= ln d) There are many interesting regular graphs with this property (including all three families of examples described in Section 2 above) Other examples appear in, e.g. 17] [3]. Another variant of Proposition 4.1 is obtained by noting that if 1 2 n are the eigenvalues of the adjacency matrix of a d regular graph G containing N cycles of length 4 then n X i=1 4 i = 8N nd 2 nd(d 1) c.f. e.g. 7] Chapter 9 for the easy argument) ....

N. Alon and M. Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs and Combinatorics 13 (1997), 217-225.

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