M. Ajtai, J. Komlos, and E. Szemeredi. A note on Ramsey numbers. Journal of Combinatorial Theory, Series A, 29: 354--360, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....l ; K n ) for odd l 5. Erd os et al. 5] proved that r(C l ; K n ) c(l)n where k = dl=2e 1; and c(l) is a positive constant depending on l. A general lower bound for r(C l ; K n ) was given by Spencer [8] Later the asymptotics of r(C 3 ; K n ) was determined up to a constant factor in [1] and [6] For other values of l the result of Erd os et al. was slightly improved by Caro et al. 4] In particular they showed that r(C 2k ; K n ) c(k) n= ln n) k= k 1) for k xed where n tends to in nity, and that r(C 5 ; K n ) cn 3=2 p ln n. In [4] the authors also Research ....

M. Ajtai, J. Komlos and E. Szemeredi, A note on Ramsey numbers, J. Combinatorial Theory A 29 (1980), 354-360.

....subjects. By the early eighties Ramsey type theorems scattered around in di#erent fields were put together to form Ramsey Theory. About the same time, capitalizing on the maturity of the subject theoretical computer science started to profit from it, initiated perhaps by Ajtai Komlos Szemeredi s [3, 4, 5] and Yao s [252] influential papers. Since then Ramsey theory has been applied in many di#erent ways in theoretical computer science and these have not been put together so far. Most of these applications are using existing theorems, but there are also papers, mostly by Alon [6, 7, 8] where new ....

....d is an (n, x, n n 1 1 d x) expanding graph for all 0 x n. This geometric expander is highly expanding (b(n) a(n) ##) with close to the smallest possible number of edges, and it is also used to obtain results for parallel sorting in rounds. 11 In 1980 Ajtai, Komlos and Szemeredi [3] proved that r(K 3 , Km ) log m) This result has been applied in the construction of algorithms to find large independent sets, see also the section on approximation algorithms) Kim (1994) 159] showed that this upper bound is tight, up to a constant factor. His argument is ....

[Article contains additional citation context not shown here]

M. Ajtai, J. Komlos and E. Szemeredi, A note on Ramsey Numbers, J. Comb. Theory Ser A 29 (1980), 354--360.

....and non neighborhood of the pivot node, before making the recursive call. If care is taken to conquer the smaller subproblem first, only linear (in n) extra space is required. For small values of k, we are able to improve on Ramsey slightly. A technique by Ajtai, Koml os, and Szemer edi [1] treated by Shearer [20] as a randomized greedy algorithm, can be made deterministic to find an independent set in k clique free graphs of in polynomial time. Performance guarantee We have seen that if the graph contains no large cliques, then Ramsey performs quite well. Unfortunately, if ....

M. Ajtai, J. Koml'os, and E. Szemer'edi. A note on Ramsey numbers. J. Combin. Theory Ser. A, 29:354--360, 1980.

....R(k; k) n. 2 Here we concentrate on l = 3 and the asymptotics as k 1. The basic upper bound, from the proof of Ramsey s Theorem, was R(3; k) Gamma k 1 which was lowered to O(k 2 ln ln k ln k ) by Graver and Yackel [12] in 1968 and then to O( ln k ) by Ajtai, Koml os and Szemer edi [1] in 1980. A lower bound R(3; k) n means that there exists a trianglefree graph G on n vertices with no independent k set. After a number of false starts a lower bound R(3; k) Omega Gamma ) was shown by Erdos [7] in 1961. This paper displays a remarkable combination of insight and ....

M. Ajtai, J. Koml'os, E. Szemer'edi, A note on Ramsey numbers, J. Combinatorial Theory (Ser A), 29 (1980), 354-360

....there relied on an overly optimistic personal communication from Spencer. Further refinements of this method are studied in [HZ2, HY] For a more in depth study of triangle free graphs in relation to the case of R (3, k ) for which considerable progress has been obtained in recent years, see also [AKS, Alon2, BBH1, BBH2, CPR, FL, Fra1, Fra2, Gri, Loc, KM1, RK3, RK4, She2, Stat, Yu1]. In 1995, Kim [Kim] obtained a breakthrough by proving that R (3, k ) has order of magnitude exactly Q(k 2 log k ) Good asymptotic bounds for R (k , k ) can be found, for example, in [Chu3, McS] lower bound) and [Tho] upper bound) and for many other asymptotic bounds in the general case ....

.... R (K 3 , G ) 2e (G ) 1 for any graph G without isolated vertices [Sid3, GK] R (K 3 , G ) n (G ) e (G ) for all G , a conjecture [Sid2] R (K 3 , K n ) see section 2 R (K 3 , K n e ) see section 3 R (K 3 , G ) for all connected G up to 9 vertices, see section 7 See [AKS, BBH1, BBH2, FL, Fra1, Fra2, Gri, Loc, KM1, RK3, RK4, She2, Stat, Yu1] 10 THE ELECTRONIC JOURNAL OF COMBINATORICS (2001) DS1.8 Paths versus other graphs: Paths versus stars [Par2, BEFRS2] Paths versus trees [FS4] Paths versus books [RS2] Paths versus cycles [FLPS, BEFRS2] Paths versus K n [Par1] Paths versus K n ,m [Hag] Paths versus W 4 and W 5 ....

M. Ajtai, J. Komlos and E. Szemeredi, A Note on Ramsey Numbers, Journal of Combinatorial Theory, Series A, 29 (1980) 354-360.

....Let f(n; be the maximum chromatic number of such a graph. The proof of Erds [4] shows that for xed , f(n; n 1= o(1) For the triangle free case = 3 this was improved by Kim [7] who proved that f(n; 3) 1 9 n 1=2 = p log n, which (by a result of Ajtai, Komls, and Szemerdi [1]) is best possible up to the value of the constant factor. It was recently shown in [9] that for xed , f(n; n 1= 1) o(1) yielding the best known lower bound for 3. The best upper bound is 2 due to Kostochka (see e.g. Jensen and Toft [6] who proved that for xed 3, f(n; ....

M. Ajtai, J. Komls, and E. Szemerdi, A note on Ramsey numbers, J. Combin. Theory Ser. A 29 (1980), 354360.

.... with maximum degree Delta and independence number at most O(n log Delta= Delta) Since every forest contains an independent set of at least half its size, these graphs also have no acyclic set of size greater than O(n log Delta= Delta) Moreover, this result is asymptotically sharp since in [1, 7], it is proved that every triangle free graph on n vertices and maximum degree Delta has an independent set of size at least Omega Gamma n log Delta= Delta) 3 Proof of Theorem 1.5 In this section we complete the proof of Theorem 1.5. Proof of Theorem 1.5: We suppose that G is a minimal ....

.... n 1 a(G Gamma n;d ) n; d : lim n 1 a(T n;d ) n; Gamma d : lim n 1 a(T Gamma n;d ) n; fi d : lim n 1 a(B n;d ) n; fi Gamma d : lim n 1 a(B Gamma n;d ) n: 11 Table of Results d = 2 3 4 5 : fl d ; fl Gamma d 2 d 1 [4] 1 2 3 8 Omega log d d [1] d , Gamma d 5 8 Lem. 2.1 Lem. 2.1 3 4 Lem. 2.1 4 7 1 2 Gamma d = Theta log d d Ex. 1.3 Ex. 2.2 Ex. 2.3 Rem. 2.4 5 9 17 32 1 2 1 2(d Gamma 1) 2 fi d , fi Gamma d Cor. 4.1 Cor. 4.1 Cor. 4.1 7 12 9 16 1 2 1 b(d 1) 2 =2c Ex. 5.1 Ex. 5.2 Cor. ....

M. Ajtai, J. Koml'os and E. Szemer'edi, A note on Ramsey numbers, J. Combinatorial Theory, Ser. A 29 (1980), 354-360.

....05C15 1. Introduction Let G be a triangle free graph of order n with average degree d, and independence number #(G) There has been great interest in finding good lower bounds for #(G)intermsofd, and producing polynomial time algorithms which find large independent sets of G. In [1] and [2] Ajtai, Komlos and Szemeredi made a breakthrough in this area when they provided a polynomial algorithm to find an independent set of size at least #(G) # n log d 100d . Correspondence to Tristan Denley, Matematiska institutionen, Umeauniversitet,Umea, Sweden Email to ....

Ajtai, M.,Koml os, J. and Szemer edi, E., A note on Ramsey Numbers,J. Comb.TheorySer.A29 (1980), 354--360.

....(3) n for which (G (3) n ) 1 9 r n log n : An easy consequence of Theorem 1.1 is c(1 Gamma o(1) t 2 log t R(3; t) with c = 1=162 = 1= 2 Delta 9 2 ) where o(1) goes to 0 as t goes to infinity. We make no attempt here to find the tightest possible constants. Because it is known [1], 2] 32] also that R(3; t) 1 o(1) t 2 log t ; 1) we now know that t 2 = log t is the correct asymptotic order of magnitude of R(3; t) Also, 1) easily gives an upper bound of (G (3) n ) which, together with Corollary 1.2, yields (1 Gamma o(1) 1 9 r n log n max G (3) n ....

.... argument that has become a cornerstone of probabilistic methods in Combinatorics (see e.g. 3] or [6] Graver and Yackel [16] found an upper bound in 1968 which, in conjunction with Erdos s bound, gave c 1 t 2 (log t) 2 R(3; t) c 2 t 2 log log t log t : Ajtai, Koml os and Szemer edi [1], 2] removed the log log t factor in the upper bound, and Shearer [32] see also [33] reduced the constant and simplified the proof to obtain (1) Meanwhile, the lower bound c 1 (t= log t) 2 defied improvement although Spencer [34] Bollob as [6] Erdos, Suen and Winkler [11] and Krivelevich ....

M. Ajtai, J. Koml'os, and E. Szemer'edi. A note on Ramsey numbers. J. of Combinatorial Th. (A), 29:354--360, 1980.

....using the method from the paper [HZ1] because the bounds there relied on an overly optimistic personal communication from Spencer. For a more in depth study of triangle free graphs in relation to the case of R (3, k ) for which considerable progress has been obtained in recent years, see also [AKS, FL, Fra1, Fra2, Gri, Loc, KM1, RK3, RK4, She2, Stat, Yu1]. In 1995, Kim [Kim] obtained a breakthrough by proving that R (3, k ) has order of magnitude exactly Q(k 2 log k ) Good asymptotic bounds for R (k , k ) can be found, for example, in [Chu3, McS] lower bound) and [Tho] upper bound) and for many other asymptotic bounds in the general case ....

.... R (K 3 , G ) 2e (G ) 1 for any graph G without isolated vertices [Sid3, GK] R (K 3 , G ) n (G ) e (G ) for all G , a conjecture [Sid2] R (K 3 , K n ) see section 2 R (K 3 , K n e ) see section 3 R (K 3 , G ) for all connected G up to 9 vertices, see section 7 See also [AKS, FL, Fra1, Fra2, Gri, Loc, KM1, RK3, RK4, She2, Stat, Yu1] THE ELECTRONIC JOURNAL OF COMBINATORICS 1 (1999) DS1 Paths versus other graphs: Paths versus stars [Par2, BEFRS2] Paths versus books [RS2] Paths versus cycles [FLPS, BEFRS2] Paths versus K n [Par1] Paths versus K n ,m [Hag] Paths and cycles versus trees [FSS] Sparse graphs versus paths and ....

M. Ajtai, J. Komlos and E. Szemeredi, A Note on Ramsey Numbers, Journal of Combinatorial Theory, Series A, 29 (1980) 354-360.

....bound to 5=14 for the case k = 3, and Fajtlowicz [5a] pro 2 vided a nonplanar graph achieving the bound of 5=14. Griggs [9] proved that if G is triangle free, then ff(G) n= k(k 1) P (1 d(v) Gamma1 , provided that G is not an odd cycle or a path. Ajtai, Koml os, and Szemer edi [0] proved that if G is a triangle free graph with average degree d , then ff(G) n ln(d ) 2:4d ) In [2] we find Question 2: Is it true that ff (G) 3=8 for every 3 regular triangle free planar graph G This is achieved by the 16 vertex planar graph below. No progress has been made, ....

M. Ajtai, J. Koml'os, and E. Szemer'edi, A note on Ramsey numbers, J. Comb. Theory (A) 29(1980), 354-360.

....number ff(G) satisfies ff(G) 0:98 e Delta 10 Gamma5=k Delta n t Delta (ln t) 1=k : 4) Various applications of Theorem 2. 2 have been found including the disproof of Heilbronn s conjecture [30] results on Sidon sets [4] Steiner systems [40] complexity theory [37] Ramsey numbers [3], geometric selection problems [34] Tur an numbers for random graphs [29] and graph coloring problems [8] 16] and [33] Indeed, for a certain range of the involved parameters k; n; t inequality (4) is best possible up to constant factors as a random hypergraph argument shows. Namely, for a ....

M. Ajtai, J. Koml'os and E. Szemer'edi, A Note on Ramsey Numbers, Journal of Combinatorial Theory, Ser. A 29, 1980, 354-360.

....Cliques are precisely the graphs that have only the trivial independent set. It is therefore natural to expect that Improving on results of Ajtai et al. [1] Schiermeyer [30] gave an algorithm that finds a large independent set in k free graphs. This generalizes results on triangle free graphs [2, 31]. Theorem 11 (Schiermeyer) There is a polynomial algorithm that finds an independent set of size Omega Gamma n log d k 2 d ) in a k clique free graph. Schiermeyer then considered a clique removal algorithm from [21] but the analysis for the approximation performance was flawed. The ....

M. Ajtai, J. Koml'os, and E. Szemer'edi. A note on Ramsey numbers. J. Combin. Theory Ser. A, 29:354--360, 1980.

....at most one loop. Hence we assume that S #n 4 and restrict ourselves to shifts s # S. Suppose that for some s # S the graph G s has less than n 24 triangles. Then there is a set A ## # A # , A ## #n 8, which induces a triangle free graph with no loops. Ajtai, Komlos, and Szemeredi [1] proved that a triangle free graph of degree at most d contains an independent set of size at least t # A ## log d 100d = n log d 800d . If there is an independent set K in G s , it follows that a bwhenever (x a s ,y b )#G for a, b # K. Hence the graph G restricted to the nodes ....

M. Ajtai, J. Koml os, and E. Szemer edi, A note on Ramsey numbers, J. Combin. Theory Ser. A, 29 (1980), pp. 354--360.

....to at most one loop. Hence we assume that jSj n=4 and restrict ourselves to shifts s 2 S. Suppose that for some s 2 S the graph G s has less than n=24 triangles. Then there is a set A 00 A 0 , jA 00 j n=8 which induces a triangle free graph with no loops. Ajtai, Koml os, Szemer edi [1] proved that a triangle free graph of degree at most d contains an independent set of size at least t jA 00 j log d 100d = n log d 800d : If there is an independent set K in G s , it follows that a b whenever (x a s ; y b ) 2 G for a; b 2 K. Hence the graph G restricted to the nodes ....

M. Ajtai, J. Koml' os, and E. Szemer' edi, A note on ramsey numbers, J. Comb. Theor. (A), 29 (1980), pp. 354--360.

....05C15 x1. Introduction Let G be a triangle free graph of order n with average degree d, and independence number ff(G) There has been great interest in finding good lower bounds for ff(G) in terms of d, and producing polynomial time algorithms which find large independent sets of G. In [1] and [2] Ajtai, Koml os and Szemer edi made a breakthrough in this area when they provided a polynomial algorithm to find an independent set of size at least ff(G) n log d 100d : Correspondence to Tristan Denley, Matematiska institutionen, Umea universitet, Umea, Sweden Email to ....

Ajtai, M.,Koml' os, J. and Szemer' edi, E., A note on Ramsey Numbers, J. Comb. Theory Ser. A 29 (1980), 354--360.

....using the method from the paper [HZ] because the bounds there relied on an overly optimistic personal communication from Spencer. For a more in depth study of triangle free graphs in relation to the case of R (3, k ) for which considerable progress has been obtained in recent years, see also [AKS, FL, Fra1, Fra2, Gri, Loc, KM, RK3, RK4, S2, Stat, Yu1]. In 1995, Kim [Kim] obtained a breakthrough by proving that R (3, k ) has order of magnitude exactly Q(k 2 log k ) Good asymptotic bounds THE ELECTRONIC JOURNAL OF COMBINATORICS 1 (1994) DS1 for R (k , k ) can be found, for example, in [Chu3, McS] lower bound) and [Tho] upper bound) ....

.... R (K 3 , G ) 2e (G ) 1 for any graph G without isolated vertices [Sid3] R (K 3 , G ) n (G ) e (G ) for all G , a conjecture [Sid2] R (K 3 , K n ) see section 2 R (K 3 , K n e ) see section 3 R (K 3 , G ) for all connected G up to 9 vertices, see section 7 See also [AKS, FL, Fra1, Fra2, Gri, Loc, KM, RK3, RK4, S2, Stat, Yu1] THE ELECTRONIC JOURNAL OF COMBINATORICS 1 (1994) DS1 Cycles versus complete graphs: R (C 4 , K 3 ) R (C 4 , C 3 ) 7 R (C 4 , K 4 ) 10 [CH2] R (C 4 , K 5 ) 14 [GG] He2, LRZ] R (C 4 , K 6 ) 18 [Ex9] RoJa1] 21 R (C 4 , K 7 ) 22 [JR1] R (C 5 , K 3 ) R ....

M. Ajtai, J. Komlos and E. Szemeredi, A Note on Ramsey Numbers, Journal of Combinatorial Theory, Series A, 29 (1980) 354-360.

....by the following theorem and proof that are due to Tuza, Tu91] Theorem 1.5 Triangle free graphs, and hence diagrams, admit colorings with O( p n log log n ) colors. Proof: First we combine two bounds on the size of independent sets in a triangle free graph. According to Ajtai et al. [AKS80], see also [Gr83] a triangle free graph (e.g. a diagram) of average degree d contains an independent set of size Omega Gamma n ln d d ) On the other hand it contains an independent set of size d, namely the neighborhood of some vertex. A simple calculation using the threshold d = p n log n ....

M. Ajtai, J. Koml' os and E. Szemer' edi, A note on Ramsey numbers, J. Comb. Th. (A) 29 (1980), 354--360.

No context found.

M. Ajtai, J. Komlos, and E. Szemeredi. A note on Ramsey numbers. Journal of Combinatorial Theory, Series A, 29: 354--360, 1980.

No context found.

Ajtai, M., Koml os, J., and Szemer edi, E. A note on Ramsey numbers. J. Combin. Theory Ser. A 29, 3 (1980), 354-360.

No context found.

M. Ajtai, J. Komlos and E. Szemeredi, A Note on Ramsey Numbers, Journal of Combinatorial Theory, Series A, 29 (1980) 354-360.

No context found.

M. Ajtai, J. Komlos and E. Szemeredi, A note on Ramsey numbers, J. Combinatorial Theory A 29 (1980), 354-360.

No context found.

M. Ajtai, J. Koml' os and E. Szemer' edi, A note on Ramsey numbers, J. Comb. Th. (A) 29 (1980), 354--360.

No context found.

M. Ajtai, J. Komlos, and E. Szemeredi, A Note on Ramsey Numbers, Journal of Combinatorial Theory, Series A 29 (1980) 354-360.

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