The independent neighborhoods process

Abstract

A triangle T (r) in an r-uniform hypergraph is a set of r + 1 edges such that r of them share a common (r − 1)-set of vertices and the last edge contains the remaining vertex from each of the first r edges. Our main result is that the random greedy triangle-free process on n points terminates in an r-uniform hypergraph with independence number O((n log n) 1/r ). As a consequence, using recent results on independent sets in hypergraphs, the Ramsey number r(T (r) , K (r) s ) has order of magnitude s r / log s. This answers questions posed in [4, 10] and generalizes the celebrated results of Ajtai-Komlós-Szemerédi [1] and Kim [9] to hypergraphs.The upper bound was proved by Ajtai-Komlós-Szemerédi [1] as one of the first applications of the semi-random method in combinatorics (simpler proofs now exist due to Shearer [12,13]). The lower bound, due to Kim [9], was also achieved by using the semi-random or nibble method. More recently, the first author [3] showed that a lower bound for r(K 3 , K s ) could also be obtained by the triangle-free process, which is a random greedy algorithm. This settled a question of Spencer on the independence number of the triangle-free process. Still more recently, Bohman-Keevash [6] and Fiz Pontiveros-Griffiths-Morris [8] have analyzed the triangle-free process more carefully and improved the constants obtained so that the gap between the upper and lower bounds for r(K 3 , K s ) is now asymptotically a multiplicative factor of 4.Given the difficulty of these basic questions in graph Ramsey theory, one would expect that the corresponding questions for hypergraphs are hopeless. This is not always the case. Hypergraphs

On the Random Greedy $F$-Free Hypergraph Process

“…Much less is known about the process when F is a k-uniform hypergraph and k ≥ 3. The only known upper bound is due to Bohman, Mubayi, and Picollelli [6], who studied the F -free process when F is a k-uniform generalization of a graph triangle (with an application to certain Ramsey numbers). In this paper, we obtain a generalization of the upper bound in [11] to strictly k-balanced hypergraphs.…”

“…These clusters form the vertices of G. Draw an edge between two vertices in G if the corresponding clusters share a hyperedge. We will use that (6) |G| ≤ (Δ(G) + 1)α(G) (which holds for all graphs) to bound the number of vertices in G. We will show that with sufficiently high probability |G| < Y 1 . (This in turn implies that at least one element of S (H n,p ) will remain in H − n,p , i.e., X − > 0.)…”

On the random greedy F -free hypergraph process

“…These clusters form the vertices of G. Draw an edge between two vertices in G if the corresponding clusters share a hyperedge. We will use that (6) |G| ≤ (∆(G) + 1)α(G) (which holds for all graphs) to bound the number of vertices in G. We will show that with sufficiently high probability |G| < Y 1 . (This in turn implies that at least one element of S ′ (H n,p ) will remain in H − n,p , i.e.…”

“…Much less is known about the process when F is a k-uniform hypergraph and k ≥ 3. The only known upper bound is due to Bohman, Mubayi and Picollelli [6], who studied the F -free process when F is a k-uniform generalisation of a graph triangle (with an application to certain Ramsey numbers). In this paper, we obtain a generalisation of the upper bound in [11] to strictly k-balanced hypergraphs.…”

The Bipartite $K_{2,2}$-Free Process and Bipartite Ramsey Number $b(2, t)$

“…Bohman and Keevash [5] have the most general results for the H-free process; they analyze the process and bound the independence number of the resulting graph for a large class of graphs H including cycles of any length as well as cliques of any size (but also all strictly 2-balanced graphs), establishing new lower bounds on Ramsey numbers R(H, K t ) where H is any fixed cycle or clique and t → ∞. Bohman, Picollelli and Mubayi [7] studied the H-free process for certain hypergraphs H, resulting in new lower bounds for the corresponding hypergraph Ramsey numbers.…”