2024/08/16 by Sankar, Maya
#05C36 #05C65 #05E45 (Primary) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2408.09029
The topological Turán number ex\hom(n,X) of a 2-dimensional simplicial complex X asks for the maximum number of edges in an n-vertex 3-uniform hypergraph containing no triangulation of X as a subgraph. We prove that the Turán exponent of any such space X is at most 8/3, i.e., that ex\hom(n,X)≤ Cn8/3 for some constant C=C(X). This improves on the previous exponent of 3-1/5, due to Keevash, Long, Narayanan, and Scott. Additionally, we present new streamlined proofs of the asymptotically tight upper bounds for the topological Turán numbers of the torus and real projective plane, which can be used to derive asymptotically tight upper bounds for all surfaces.