For k ≥ 2 and r ≥ 1 such that k + r ≥ 4, we prove that, for any > 0, there exists > 0 such that the union of an n-vertex k-graph with minimum codegree

Erdős conjectured that every n‐vertex triangle‐free graph contains a subset of ⌊n/2⌋ vertices that spans at most n2/50 edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so‐called Andrásfai graphs. As a consequence, Erdős' conjecture holds for every triangle‐free graph G with minimum degree δ(G)>10n/29 and if χ(G)≤3 the degree condition can be relaxed to δ(G)>n/3. In fact, we obtain a more general result for graphs of higher odd‐girth.

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Abstract. Erdős conjectured that every n-vertex triangle-free graph contains a subset of tn{2u vertices that spans at most n 2 {50 edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so-called Andrásfai graphs. As a consequence, Erdős' conjecture holds for every triangle-free graph G with minimum degree δpGq ą 10n{29 and if χpGq ď 3 the degree condition can be relaxed to δpGq ą n{3. In fact, we obtain a more general result for graphs of higher odd-girth. §1. IntroductionWe say an n-vertex graph G is pα, βq-dense if every subset of tαnu vertices spans more than βn 2 edges. Given α P p0, 1s Erdős, Faudree, Rousseau, and Schelp We say a graph G is a blow-up of some graph F , if there exists a graph homomorphism ϕ : G Ñ F with the property tx, yu P EpGq ðñ tϕpxq, ϕpyqu P EpF q. Moreover, a blow-up is balanced if the preimages ϕ´1pvq of all vertices v P V pF q have the same size.

We show that every 3-uniform hypergraph H " pV, Eq with |V pHq| " n and minimum pair degree at least p4{5`op1qqn contains a squared Hamiltonian cycle.This may be regarded as a first step towards a hypergraph version of the Pósa-Seymour conjecture.2010 Mathematics Subject Classification. Primary: 05C65. Secondary: 05C45.