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Agraïments: This  research  was  supported  through  the  programme  “Research  in  Pairs”  by the Mathematisches Forschungsinstitut Oberwolfach in 2013.  Alex Küronya was partially supported by the DFG-Forschergruppe 790 ”Classification of Algebraic Surfaces  and  Compact  Complex  Manifolds“,  by  the  DFG-Graduiertenkolleg 1821 ”Cohomological Methods in Geometry“, and by the OTKA grants 77476 and  81203  of  the  Hungarian  Academy  of  Sciences.   Brian  Harbourne’s  work on this project was partially sponsored by the National Security Agency under Grant/Cooperative agreement “Advances on Fat Points and Symbolic Powers,” Number H98230-11-1-0139.  The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notice. 245 10It is well known that multi-point Seshadri constants for a small number t of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for t≥9 points. Tackling the problem in the language of valuations one can make sense of t points for any real t≥1. We show somewhat surprisingly that a Nagata-type conjecture should be valid for t≥8+1/36 points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for t≤7+1/9. In the range 7+1/9≤t≤8+1/36 we are able to compute some sporadic values

Very general monomial valuations of $\mathbb{P}^2$ and a Nagata-type conjecture