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 linear nvertex 3-uniform hypergraph which does not contain a b-bounded even subgraph. In this paper, we show that if b ≥ 12, then n logn 3b log logn ≤ fb(n) ≤ Bn(logn) 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 cn logn ≤ f2(n) ≤ Cn(logn). We conjecture that f2(n) = n 1+o(1) as n→∞.