2024/12/11 by Balogh, József, Halfpap, Anastasia, Lidický, Bernard +1 · 1 citation
#05C35 #05C65 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.08597
The minimum positive co-degree of a nonempty r-graph H, denoted by δr-1+(H), is the largest integer k such that for every (r-1)-set S ⊂ V(H), if S is contained in a hyperedge of H, then S is contained in at least k hyperedges of H. Given a family F of r-graphs, the positive co-degree Turán function co+ex(n,F) is the maximum of δr-1+(H) over all n-vertex r-graphs H containing no member of F. The positive co-degree density of F is γ+(F) = \undersetn → ∞lim (co+ex(n,F))/(n). While the existence of γ+(F) is proved for all families F, only few positive co-degree densities are known exactly. For a fixed r ≥ 2, we call α∈ [0,1] an achievable value if there exists a family of r-graphs F with γ+(F) = α, and call α a jump if for some δ> 0, there is no family F with γ+(F) ∈ (α, α+ δ). Halfpap, Lemons, and Palmer showed that every α∈ [0, (1)/(r)) is a jump. We extend this result by showing that every α∈ [0, (2)/(2r -1)) is a jump. We also show that for r = 3, the set of achievable values is infinite, more precisely, (k-2)/(2k-3) for every k ≥ 4 is achievable. Finally, we determine two additional achievable values for r=3 using flag algebra calculations.