{"paper":{"title":"Generalizations of the Szemer\\'edi-Trotter Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Yang, Micha Sharir, Noam Solomon, Saarik Kalia","submitted_at":"2014-08-25T20:12:54Z","abstract_excerpt":"We generalize the Szemer\\'edi-Trotter incidence theorem, to bound the number of complete \\emph{flags} in higher dimensions. Specifically, for each $i=0,1,\\ldots,d-1$, we are given a finite set $S_i$ of $i$-flats in $\\R^d$ or in $\\C^d$, and a (complete) flag is a tuple $(f_0,f_1,\\ldots,f_{d-1})$, where $f_i\\in S_i$ for each $i$ and $f_i\\subset f_{i+1}$ for each $i=0,1,\\ldots,d-2$. Our main result is an upper bound on the number of flags which is tight in the worst case.\n  We also study several other kinds of incidence problems, including (i) incidences between points and lines in $\\R^3$ such th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5915","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}