As a higher analogue of the edge ideal of a graph, we study the t-connected ideal \({{\,\textrm{J}\,}}_{t}\) . This is the monomial ideal generated by the connected subsets of size t. For chordal graphs, we show that \({{\,\textrm{J}\,}}_{t}\) has a linear resolution iff the graph is t-gap-free, and that this is equivalent to having linear quotients. We then show that if G is any gap-free and t-claw-free graph, then \({{\,\textrm{J}\,}}_{t}(G)\) has linear quotients and, hence, linear resolution.
