{
  "id": "2110.12314",
  "version": "v3",
  "published": "2021-10-23T22:59:48.000Z",
  "updated": "2022-03-24T16:40:21.000Z",
  "title": "Simplicial complexes from finite projective planes and colored configurations",
  "authors": [
    "Matt Superdock"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as $T_{1} \\sqcup T_{2}$, such that each $T_{i}$ represents the lines of a copy of the Fano plane $PG(2, \\mathbb{F}_{2})$. We generalize this observation by constructing, for each prime power $q$, a simplicial complex $X_{q}$ with $q^{2} + q + 1$ vertices and $2(q^{2} + q + 1)$ facets consisting of two copies of $PG(2, \\mathbb{F}_{q})$. Our construction works for any \\emph{colored $k$-configuration}, defined as a $k$-configuration whose associated bipartite graph $G$ is connected and has a $k$-edge coloring $\\chi \\colon E(G) \\to [k]$, such that for all $v \\in V(G)$, $a, b, c \\in [k]$, following edges of colors $a, b, c, a, b, c$ from $v$ brings us back to $v$. We give one-to-one correspondences between (1) Sidon sets of order 2 and size $k + 1$ in groups with order $n$, (2) linear codes with radius 1 and index $n$ in $A_{k}$, and (3) colored $(k + 1)$-configurations with $n$ points and $n$ lines. As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-03-24T16:40:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "05B10",
      "05B25",
      "05B30",
      "05C15",
      "05E18",
      "11T71",
      "12K10",
      "51E15",
      "51E30",
      "94B25"
    ],
    "keywords": [
      "finite projective planes",
      "simplicial complex",
      "colored configurations",
      "sidon sets",
      "planar difference sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}