Air traffic flow management in Europe has to deal as much with capacity constraints in en route airspace as with the more usual capacity constraints at airports. The en route sector capacity constraints, in turn, generate complex interactions among traffic flows. We present a deterministic optimization model for the European air traffic flow management (ATFM) problem. The model designs flow management strategies involving combinations of ground and airborne holding. The paper illustrates the complex nature of European (EU) ATFM solutions, the benefits that can be obtained by purposely assigning airborne holding delays to some flights, and the issues of equity that arise as a result of the interactions among traffic flows. In particular, we show that, in certain circumstances, it is better, in terms of total delay and delay cost, to assign to a flight a more expensive airborne holding delay than a ground delay. We also show that in the EU ATFM context, fundamental conflicts may often arise between the objectives of efficiency and equity (or "fairness"). This finding may have profound implications for the possibility of developing a "collaborative decision-making" environment for air traffic flow management in Europe.