Why a contested Republican convention will not produce a third-party challenge
There has been some discussion— including by Donald Trump himself recently—that if one Republican candidate or another loses the nomination at a contested convention in Cleveland, then that person might "bolt" and run as a third-party candidate. This scenario is most frequently invoked with respect to Donald Trump if he has a plurality of delegates but is nonetheless deprived the nomination. This scenario is highly unlikely.
The problem is one of state ballot access laws, which provide different filing deadlines and signature thresholds from one state to another. The Republican convention is in Cleveland from July 18 to July 21. Yet according to this list on Ballotpedia (I assume it is accurate), 12 states have deadlines for ballot access on or before July 21 (including several large states such as Florida, Georgia, Michigan and Texas). Another 14 states have deadlines that fall between the end of the convention and Aug. 2, during which a candidate would be required to assemble a team to acquire thousands or in some cases tens of thousands of signatures (in general, however, most of these states seem to have relatively low signature thresholds). Twenty-five state deadlines fall after Aug. 2, but many are still in early August.
Whatever happens in Cleveland, one thing thus seems almost certain—it will not produce a spontaneous third-party candidacy for president. If one of the Republican field is thinking of running as a third-party candidate, they would need to decide to do so before the Cleveland convention, which would also almost certainly doom their prospects of being the Republican nominee in the interim. In some cases they would be required to do so well before the convention and before the primaries even end. There are some news reports that some Republicans are considering doing that, which would require deciding well before the Republican candidate is finally identified.
This doesn't rule out a write-in candidacy, of course (which would be much more difficult). Nor does it rule out that possibility that those who supported a defeated candidate might simply choose not to vote or might vote for the Democratic nominee. This analysis leaves aside so-called "sore loser" laws (which Ballotpedia claims are rarely invoked in presidential elections). But the timing simply doesn't work for a third-party challenge.