Homomorphism Thresholds for Odd Cycles

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph F we define the homomorphism threshold as the infimum over all α ∈ [0,1] such that every n -vertex F -free graph G with minimum degree at least αn has a homomorphic image H of bounded order (i.e. independent of n ), which is F -free as well. Without the restriction of H being F -free we recover the definition of the chromatic threshold, which was determined for every graph F by Allen et al. [1]. The homomorphism threshold is less understood and we address the problem for odd cycles.

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