Asymmetric Ramsey properties of random graphs involving cliques

Abstract

Consider the following problem: For given graphs G and F1,…,Fk, find a coloring of the edges of G with k colors such that G does not contain Fi in color i. Rödl and Ruciński studied this problem for the random graph Gn,p in the symmetric case when k is fixed and F1 = ··· = Fk = F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p ≤ bn−β for some constants b = b(F,k) and β = β(F). This result is essentially best possible because for p ≥ Bn−β, where B = B(F,k) is a large constant, such an edge‐coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n for arbitrary F1,…,Fk.In this article we address the case when F1,…,Fk are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k‐edge‐coloring of Gn,p with p ≤ bn−β for some constant b = b(F1,…,Fk), where β = β(F1,…,Fk) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F1,…,Fk) such that for p ≥ Bn−β the random graph Gn,p a.a.s. does not have a valid k‐edge‐coloring provided the so‐called KŁR‐conjecture holds. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Independent sets in hypergraphs

“…More accurately, the above conjecture was stated in [39] only in the case r = 2, but the above generalization is quite natural, see [46]. There had been little progress on Conjecture 1.10 until quite recently, when the 0-statement was proved by Marciniszyn, Skokan, Spöhel, and Steger [46] in the case where all of the H i are cliques, and the 1-statement in the case r = 2 was established by Kohayakawa, Schacht, and Spöhel [44] under very mild extra assumptions on H 1 and H 2 .…”

Extremal Results in Random Graphs

“…In [62] the 1-statement of Conjecture 4.1 for R n (C, F ) for any cycle C and any 2-balanced graph F with m 2 (C) ≥ m 2 (F ) was verified. Moreover, the 0-statement was shown for the case when F 1 and F 2 are cliques [82], and the 1-statement was shown for graphs F 1 and F 2 with m 2 (F 1 , F 2 ) > m 2 (F 1 , F ) for every F F 2 with e(F ) ≥ 1 appeared in [69]. In particular, those results yield the threshold for R(K k , K ).…”

“…It was also known that the resolution of Conjecture 3.6 for the (sparser) graph F 1 allows us to generalize the proof from [62] to verify the 1-statement of Conjecture 4.1 when F 2 is strictly 2-balanced (see, e.g., [82]). Therefore, Theorem 3.7 has the following consequence.…”

“…For example, following the proof from [62] (see also [82]), one can use the positive resolution of Conjecture 3.6 to prove the 1-statement of the threshold for the asymmetric Ramsey properties of random graphs (see Section 4.1), but Theorems 3.8 and 3.9 seem to be insufficient for this application. In Section 4 we will also mention some applications, which require the quantitative estimates of Theorem 3.9 (see Section 4.3 and 4.4).…”

Upper bounds on probability thresholds for asymmetric Ramsey properties

“…The approach via sparse regularity can be extended to prove the 1‐statement of Conjecture 3 for any two graphs G and H , provided the so‐called KŁR‐Conjecture holds for G and H is strictly balanced w.r.t. (see ; additionally, Lemma 16 in needs to be modified slightly to relax the condition on H from ‘2‐balanced’ to ‘strictly balanced w.r.t. ’).…”