Some older books I've seen say that the minimum number of stages of an explicit Runge-Kutta method of a specified order is unknown for orders $\geq 9$. Is this still true?What libraries are there for automatically working with high order Runge-Kutta methods?
Constructing explicit Runge Kutta methods of order 9 and higher
ode;runge kutta
BoundsThat is still true.  In Butcher's book, page 196, it says the following: In a 1985 paper, Butcher showed that you need 11 stages to get order 8, and this is sharp.  For order 10, Hairer derived a family of 17-stage methods, but it's not known if one can do better.  The same information is given in Section II.5 of Hairer, Norsett, & Wanner vol. I.  The latter reference also goes through some of the techniques for developing high-order pairs (up to order 8).There is an upper bound on the minimum number of stages required for any order, since you can construct them by extrapolation.  This has been known for a very long time; see this recent paper of mine for an explanation. However, this bound is quadratic in the order and is certainly quite pessimistic.  The nodepy software discussed below can generate exact coefficients for these methods, and also for deferred correction methods (which are Runge-Kutta methods) of any order.I believe @Etienne is correct in saying that the highest order methods that have been constructed by hand are due to Terry Feagin.  Regarding his other comment, this paper contains some 9(8) pairs:J.H. Verner, High-order explicit Runge-Kutta pairs with low stage order, Applied Numerical Mathematics, Volume 22, Issues 13, November 1996, Pages 345-357Here is a table of the (cumulative) number of order conditions $N$ required for each order $p$; this table goes further than those provided in the literature and was produced using nodepy:p | N-----1 | 12 | 23 | 44 | 85 | 176 | 377 | 858 | 2009 | 48610| 120511| 304712| 781313| 2030014| 53264SoftwareFor very high order methods, the number and complexity of the order conditions becomes impossible to deal with by hand.  Some symbolic packages (Mathematica, at least) have capabilities for generating Runge-Kutta order conditions.  There are probably some other packages out there, but I am aware of the following (both of which I wrote):nodepy: a Python package that can generate symbolic expressions and code for the order conditions to arbitrary order.  It also includes Python code for checking those conditions, computing error coefficients, etc.RK-Opt: A MATLAB package that, among many other things, can find high order Runge-Kutta methods with coefficients optimized for a few different purposes.  It currently couldn't handle 9th-order explicit RK (it goes up to 8th order for stage-order-one methods, tenth order for methods with higher stage order).  If that's something you're interested in I could add the 9th-order (and higher) conditions fairly easily.Another interesting note about order conditions, which becomes significant at such high orders, is that there are two ways to derive them, and they give you different (but collectively equivalent) conditions: one is due to Butcher, the other to Albrecht.