Large graphs are very hard to analyze; random graphs are one method
trying to overcome this problem. We introduce several models of random
graphs, describe a formal first-order language characterizing such
graphs, and discuss its limitations. We believe, though, that in a
way, first-order logic is more "natural", hence it is interesting
enough to get familiar with the not so obvious result regarding the
"Zero-One Law", stating that for a large enough family of edge-density
functions first-order sentences hold almost always or almost never. To
prove this we use a blend of combinatorics, logic, probability and
popular culture. If time allows, we will introduce a nice (though not
so popular) game played on graphs, that as a bonus allows us to make
sure a second-order sentence is not a first-order sentence in
disguise.