On Even-Degree Subgraphs of Linear Hypergraphs

A subgraph of a hypergraph H is even if all its degrees are positive even integers, and b-bounded if it has maximum degree at most b. Let fb(n) denote the maximum number of edges in a linearn-vertex 3-uniform hypergraph which does not contain a b-bounded even subgraph. In this paper, we show that if b ≥ 12, then \[ \frac{n \log n}{3 b\log \log n} \leq f_b(n) \leq Bn(\log n)^2 \] for some absolute constant B, thus establishing fb(n) up to polylogarithmic factors. This leaves open the interesting case b = 2, which is the case of 2-regular subgraphs. We are able to show for some constants c, C > 0 that \[ c n\log n \leq f_2(n) \leq Cn^{3/2}(\log n)^5. \] We conjecture that f2(n) = n1 + o(1) as n → ∞.