Both words refer to the most important of a set of things. Main is very common, and can suggest not only the most important but also the biggest or most extensive: the main highway; the main reason. Main can attach to a plural—the main sources of energy, whereas paramount is only applied to the one thing that is of chief importance or concern, above all others: It is an issue of paramount importance; Safety is paramount. Paramount is a much stronger term than main, not only because it refers to a higher degree of importance, but also because it suggests something critical or essential that needs tending to.
It’s not difficult to separate these two words. Everyone knows the affectionate embrace of hugging someone. The word possibly comes from an Old Norse word, hugga, meaning to soothe or console. Clinch suggests tightly holding someone for other purposes—usually to prevent them from moving. Boxers clinch their opponents by putting their arms around the opponent’s body and arms, and leaning on them to prevent punches or to gain time until one of them manages to break out of the hold. This verb is usually used intransitively: Sugar Ray Robinson and his opponent clinched. The more frequent use of clinch is as a verb meaning to secure a deal, title, or victory—that is, to make sure it’s not going anywhere.
Both words refer to growing or becoming greater. To increase is to grow in number, size, strength, or quality. It implies the simple fact of something becoming greater as opposed to shrinking or staying the same: The population increased by 2 percent. Escalate, which derives from escalator, an invention in use by the 1920s, suggests a steeper, sometimes alarming increase in intensity or magnitude, sometimes in prices, but more often in tense situations. If a situation, confrontation, or conflict escalates, it begins to get out of hand or threatens to escalate into something full-blown.