On a Ramsey--Turán Variant of the Hajnal--Szemerédi Theorem

Abstract

A seminal result of Hajnal and Szemerédi states that if a graph G with n vertices has minimum degree δ(G) ≥ (r − 1)n/r for some integer r ≥ 2, then G contains a K r -factor, assuming r divides n. Extremal examples which show optimality of the bound on δ(G) are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Túran theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results:• For any r > ℓ ≥ 2, if G is a graph satisfying δ(G) ≥ r−ℓ r−ℓ+1 n+Ω(n) and α ℓ (G) = o(n), that is, a largest K ℓ -free induced subgraph has at most o(n) vertices, then G contains a K r -factor. This is optimal for ℓ = r − 1 and extends a result of Balogh, Molla, and Sharifzadeh who considered the case r = 3.• If a graph G satisfies δ(G) = Ω(n) and α * r (G) = o(n), that is, every induced K r -free r-partite subgraph of G has at least one vertex class of size o(n), then it contains a K r -factor. A similar statement is proven for a general graph

Hamiltonicity in randomly perturbed hypergraphs

“…There exists a bipartite graph B with vertex classes X m ∪ Y m and Z m and maximum degree ∆(B) ≤ 40, such that |X m | = m + βm, |Y m | = 2m and |Z m | = 3m, and for every subsetX ′ m ⊆ X m with |X ′ m | = m, the induced graph B[X ′ m ∪ Y m , Z m ] contains a perfect matching.We use Lemma 4.2 to prove the following absorbing lemma for hypergraphs, obtaining a sufficient condition for the existence of an absorbing set. This lemma extends to k-graphs a result for graphs (k = 2) obtained in[31, Lemma 2.2].…”

“…Let us first give the following definition concerning the absorption method. In order to obtain an absorbing set, we need the following result from [31], which follows from [30,Lemma 10.7] (see also [29,Lemma 2.8]). Define the constants q = γ/(500b 2 ) and β = q b−1 γ/4, and put ξ = βq/(2(1 + β)(b − 1)).…”

Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

“…It is not difficult to prove the existence of such templates for large enough m probabilistically; see, for example, [42,Lemma 2.8] . The idea has since be used by various authors in different settings [16,17,20,21,39,44]. We will use a template here as an auxiliary graph in order to build an absorbing structure for our purposes.…”

“…Additional note : Since the paper was first submitted there have been some related results proven. Indeed, Knierim and Su [30], expanding on work of Nenadov and Pehova [44], considered the so‐called Ramsey‐Turán problem for clique tilings. They showed that for any , there exists an such that if G′ is a graph with and independence number less than , then G′ contains a perfect K r ‐tiling.…”

Finding any given 2‐factor in sparse pseudorandom graphs efficiently

“…Kwan [40] also used sparse templates to study random Steiner triple systems, generalising the template to a hypergraph setting and using it to define an absorbing structure for perfect matchings. Further applications were given by Ferber and Nenadov [23] in their work on universality in the random graph, recently by the current authors in [24] which was the first use of the method in the context of pseudorandom graphs, and by Nenadov and Pehova [45] who used the method to study a variant of the Hajnal‐Szeméredi Theorem. The final three papers mentioned all adapt the method to give absorbing structures which output disjoint copies of a fixed graph (a partial ‐factor), however the different absorbing structures used are interestingly all significantly distinct.…”