The structure of Deitmar schemes, I

We explain how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, F-1) to a so-called "loose graph" (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and it also appears that known realizations of objects over F-1 (such as combinatorial F-1-projective and F-1-affine spaces) exactly depict the loose graph which corresponds to the associated Deitmar scheme. This idea is then conjecturally generalized so as to describe all Deitmar schemes in a similar synthetic manner.We explain how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, F-1) to a so-called "loose graph" (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and it also appears that known realizations of objects over F-1 (such as combinatorial F-1-projective and F-1-affine spaces) exactly depict the loose graph which corresponds to the associated Deitmar scheme. This idea is then conjecturally generalized so as to describe all Deitmar schemes in a similar synthetic manner.A