Short proofs for long induced paths

Abstract

We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies $\hat{R}_{\mathrm{ind}}(P_n)\leq 5 \cdot 10^7n$ , thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and Łuczak. We also provide a bound for the k-colour version, showing that $\hat{R}_{\mathrm{ind}}^k(P_n)=O(k^3\log^4k)n$ . Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, $G(n,\frac{1+\varepsilon}{n})$ , contains typically an induced path of length $\Theta(\varepsilon^2) n$ .

Effective bounds for induced size-Ramsey numbers of cycles

“…For paths it is known that Ω(k 2 )n ≤ rk (P n ) ≤ O(k 2 log k)n (see [10,24] for the lower bound and [9,25] for the upper bound). In the induced case, by a recent result of Draganić, Krivelevich and Glock [6], we have that rk ind (P n ) ≤ O(k 3 log 4 k)n. For cycles, the discrepancy between the size-Ramsey and the induced size-Ramsey number is significantly larger. Indeed, by a recent result of Javadi and Miralaei [20], which improved another recent result by Javadi, Khoeini, Omidi and Pokrovskiy [19], we have r k (C n ) = O(k 120 log 2 k)n for even n, and r k (C n ) = O(2 16k 2 +2 log k )n for odd n. On the other hand, the only known upper bound on the induced size-Ramsey numbers of cycles was obtained in the seminal paper of Haxell, Kohayakawa and Łuczak [17].…”

“…Further, we also know that it is locally sparse, that is, all sets U of size |U | ≤ εN span at most 3 2 |U | edges, where ε > 0 is some constant depending on C. We consider the subgraph corresponding to the densest colour class, say red and using a result of Krivelevich [24], we find inside it a large expanding subgraph G . Draganić, Glock and Krivelevich [6] showed using a modified DFS algorithm that under the given assumptions, G has a red induced path of length 2n/5 and we adapt their argument to our setting. Given such a red induced path of length 2n/5, from the endpoints we construct two trees T 1 , T 2 each of depth O(log N ) and with Ω(εN ) leaves.…”

“…Lemma 3.4. There exists a positive constant C such that for any positive integer k, there exists a graph F 0 (k) with (Ck) 6 vertices and (Ck) 9 /6 edges such that any k-edge-colouring contains a monochromatic 6-cycle forming an induced subgraph of F 0 (k). In particular, Rk ind (C 6 ) = O(k 9 ).…”

“…Clearly we may assume that k is large enough. Let n = (Ck) 6 where C is a large constant to be chosen later. Let F 0 (k) be a bipartite C 4 -free graph on n vertices with at least n 3/2 /8 edges given by Theorem 3.3.…”