On an anti-Ramsey threshold for random graphs

Abstract

For graphs G and H, let G rb ÝÑ p H denote the property that for every proper edge-colouring of G there is a rainbow H in G. It is known that, for every graph H, an asymptotic upper bound for the threshold function p rb H " p rb H pnq of this property for the random graph Gpn, pq is n´1 {m p2q pHq , where m p2q pHq denotes the so-called maximum 2-density of H. Extending a result of Nenadov, Person, Škorić, and Steger [J. Combin. Theory Ser. B 124 (2017), 1-38] we prove a matching lower bound for p rb K k for k ě 5. Furthermore, we show that p rb K4 " n´7 {15 . §1. IntroductionLet r be a positive integer and let G and H be graphs. We denote by G Ñ pHq r the property that any colouring of the edges of G with at most r colours contains a monochromatic H in G. In 1995, Rödl and Ruciński determined the threshold for the property Gpn, pq Ñ pHq r for all graphs H. The maximum 2-density m p2q pHq of a graph H is given bywhere we suppose |V pHq| ě 3.Theorem 1 (Rödl and Ruciński [8,9]). Let H be a graph containing a cycle. Then, the threshold function p H " p H pnq for the property Gpn, pq Ñ pHq r is given by p H pnq " n´1 {m p2q pHq .

On an anti‐Ramsey threshold for sparse graphs with one triangle

“…Lemma below can be proved by combining Lemmas 3.3 and 3.6 in . Lemma Let , and be given.…”

“…Note that the property ( , ) rb ← ← ← ← ← ← ← ← ← → p admits a threshold function rb = rb ( ) for any fixed graph , because it is an increasing property [2]. The following result, proved in [5], gives an upper bound for this threshold. Theorem 2.…”

“…We write if, for every proper edge‐coloring of G (with an arbitrary number of colors), there is a totally multicolored , or rainbow , copy of H in G , that is, a copy of H with no two edges of the same color. We remark that the term “anti‐Ramsey” is used in different contexts, but we follow the terminology in . Note that the property admits a threshold function for any fixed graph H , because it is an increasing property .…”

An algorithmic framework for obtaining lower bounds for random Ramsey problems extended abstract

“…However, their lower bound applies only for large enough r which depends on F , while a priori there is no reason why the result should not be true for all r ≥ 2. The question of proper colourings was introduced by Kohayakawa, Konstadinidis and Mota [6,7], where they proved upper bounds. Ramsey type problems have also been studied in a game-theoretic setting.…”

“…Only much later, Kohayakawa et. al [6,7] started a systematic study of this property in the random settings. In particular, they proved that the upper bound is as expected.…”