Abstract
In this paper, we consider the Turán problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\},\{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Turán densities of some small $\{1,3\}$-hypergraphs.
Highlights
Turan theory is an important and active area in the extremal combinatorics
We prove that a {1, 3}-hypergraph is degenerate if and only if it’s H5{1,3}-colorable, where H5{1,3} is a hypergraph with vertex set V = [5] and edge set E = {{2}, {3}, {1, 2, 4}, {1, 3, 5}, {1, 4, 5}}
In 1941, Turan [10] determined the graph with maximum number of edges among all simple graphs on n vertices that doesn’t contain the complete graph K as a sub-graph
Summary
Turan theory is an important and active area in the extremal combinatorics. In 1941, Turan [10] determined the graph with maximum number of edges among all simple graphs on n vertices that doesn’t contain the complete graph K as a sub-graph. Given a family of hypergraphs H with common set of edge types R, we say G is H-free if G doesn’t contain any member of H as a sub-graph. Lu and Johnston [4] proved that this limit always exists by a simple average argument of Katona-Nemetz-Simonovits theorem [6] They completely classified the Turan densities of {1, 2}-graphs. For the special case R = {r}, Erdos [3] showed that an r-uniform hypergraph H is degenerate if and only if it is r-partite, that is, a sub-graph of a blow-up of a single edge of cardinality r. A result following Theorem 4 indicates a break for the Turan density of {1, 3}-graphs: Corollary 5.
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