sign problems for U/t up to 7, [39], illustrate the hazards of the enterprise. While the last give the upper limit, T$_{c}$/t < 10 − $^{3}$, the various approximations give a value one and sometimes two orders of magnitude larger. Since (apart from the RPA), such calculations present results for T$_{c}$ without providing the form of the fluctuations spectra or of the coupling function or calculate normal state properties, it is hard to say what exactly goes wrong in even such elaborate and careful calculations. A hint for one of the possible deficiency from the best Monte-Carlo [39] calcu- lations done at various sizes of the lattice (up to 10 × 10) is that the nearest neighbor static spin-correlations of the d-wave superconducting wave-functions on a square lattice match those of the AFM wavefunction. So, it is possible for calculations on clusters of small size to lead to misleading conclusions which disappear for large enough size. In a later section, we will review the results of s-wave pairing in the attractive Hubbard model, where reliable results without numerical difficulties are available in special limits. Noting the prefactor in the BCS expression for T$_{c}$ , it is tempting to suggest attractive interaction among electrons near the fermi-surface through exchange of fluctuations whose frequency is high unlike the case of phonons where it is limited by M − 1 / $^{2}$, where M is the ionic mass [40–42]. (Note that there is no factor of M in the coupling constant λ .) These other excitations must then be fluctuations of electronic degrees of freedom themselves or photons. For the latter, the coupling constant λ is inevitably proportional to the fine- structure constant. So we need consider only the electronic excitations. It seems that starting with Little [40] no known excitation has not been thought of in this context. Although some of the ideas are not correct to begin with, most are in principle correct; the problem is in the smallness of the coupling constant. When they are wrong in principle, the mistake is of the following kind. Consider electron-electron interaction to second order in the Coulomb interaction: it is attractive. But if the whole series is summed, it gives just the screened repulsive Thomas-Fermi interaction. An example of this in connection with a proposal [41] was provided by Inkson and Anderson [43] and further checked through detailed calculations [44]. The original proposal by Little [40] to use the excitons of an insulating side-chain in an organic metal has similar problems. It should be noted however that essentially every e-e mechanism we can think of is an excitation in the particle-hole channel and that if we get a decent T$_{c}$ , it is generally always 26 E. Excitonic and similar Pairing