hounds, Rainsford leaps off a cliff and into the sea.

5. Rainsford makes his reappearance, revealing that he had swum around the island. He fights Zaroff and kills him. 

I agree with most of the other answer posted, with one obvious difference.

1. Rainsford falls off the yacht and swims to shore.

2. Rainsford meets Zaroff and finds out that he will be tomorrow's headline event (he'll be hunted!)

3. Rainsford is hunted over the span of a day or two.

4. Rainsford chooses to leap off the cliff (I consider this the climax, but there is much disagreement on this issue.)

5. Rainsford confronts and kills Zaroff.

While Connell's "The Most Dangerous Game" is a suspenseful tale throughout the entire narrative, perhaps, the three most suspenseful events are the following:

After Rainsford is discovered on the shore of Ship-Trap Island and brought to the chateau of General Zaroff, Rainsford learns that for his elegance and learning, the Cossack general is a sadistic man who hunts men for sport. When the general extends to Rainsford in his suave way an invitation to hunt with him, Rainsford refuses and asks to leave the island. Shrugging, the general explains that Rainsford may find his idea of sport "more diverting than Ivan's," intimating that the hunting dogs may be employed against him. "You don't mean--" Rainsford cries out in comprehension that he is to be hunted. He is, then, given clothing and helpful tips about wearing moccasins and avoiding Death Swamp.

1. At first, Rainsford tries to get as far away as possible, but then he realizes he is working in side the "picture frame" of an island. So he creates a trail meant to divert Zaroff. Finally, it grows dark and Rainsford, careful not to leave footprints, climbs a tree. 

But, morning finds the Cossack following the trail to the tree. Lighting one of his "pungent incense-like" black cigarettes Zaroff follows with his eyes the branching of the tree. "Very deliberately, he blows a smoke ring into the air; then he turned his back on the tree, and walked carelessly away." As Rainsford assesses the superior skill of Zaroff to have been able to follow through the woods an extremely challenging trail at night, he understands that the general must have seen him when he looked into the trees' branches.

The general was saving him for another day's sport! The Cossack was the cat; he was the mouse. Then it was that Rainsford knew the full meaning of terror.

2. After sliding down from the tree, Rainsford goes forward until he sees a huge, dead tree. With the branches he fashions a trap--a "Malay man-catcher." Rainsford the mouse awaits the return of the cat. Shortly, Zaroff approaches, scrutinizing every blade of grass, every bent twig, every little mark until he is almost upon the trap. "His foot touched the protruding bough that was the trigger." But, with amazing swiftness, the agile Zaroff jumped back in time, receiving only a swift blow to his shoulder. He calls out to Rainsford, congratulating him, adding that he must return to have his wound attended; however, he adds, he will be return.

After the general leaves, Rainsford takes flight, but steps into Death Swamp and its quicksand. Fortunately, he is able to dig his way out. From hard saplings he shapes stakes and makes another trap. Still, he cannot rest even in the dusk because he hears Zaroff coming. He hears a sharp scream and jumps up excitedly, but he must quickly cower back down as he has seen a man with a torch.

"Your Burmese tiger pit has claimed one of my best dogs. Again you score...Mr. Rainsford, "I'll see what you can do against my whole pack...Thank you for a most amusing evening."

3. At daybreak Rainsford learns new fears: he hears the baying of a pack of hounds. He quickly examines his options: he can stay and wait, but that could mean suicide; he could run, but that would be "postponing the inevitable." Suddenly, he thinks of running away from the swamp. When he reaches a ridge, Rainsford climbs a tree and spots the brush moving in the distance; then, he discerns Ivan with the hounds. "They would be on him any minute now. His mind worked frantically," and he remembers a native trick from Uganda. Grabbing a green sapling, Rainsford bends it down, attaching his knife to its end pointing down the trail; he ties the sapling in place with vine.

Then he ran for his life. The hounds raised their voices as they hit the fresh scent. Rainsford knew now how an animal at bay feels.

He has to stop when he is out of breath and climbs another tree to survey where his trap has been. The general is on his feet, but Ivan is not. Rainsford steels himself and runs again towards the "blue gap" of the sea. He reaches it, but the sea is twenty feet below. "He heard the hounds. Then he leaped far out into the sea."

[The final episode that is suspenseful is, of course, the appearance of Rainsford in the bedroom of General Zaroff and the duel between them.]

The rising action is a series of significant events that increase the tension of the story, introduce the conflict, and build suspense. The rising action also takes place after the exposition and ends at the climax of the story. In Connell's celebrated short story "The Most Dangerous Game," the rising action begins the moment that Sanger Rainsford falls off the yacht. After Whitney heads below deck, Rainsford stays up and suddenly hears the sound of a gunshot coming from the direction of the nearby island. Rainsford proceeds to leap onto the rail of the yacht and accidentally knocks his pipe into the sea. When Rainsford lunges for his pipe, he falls overboard and the yacht continues to sail away. At this moment, the rising action begins as the suspense immediately increases. Rainsford is in the middle of a completely black sea and must brave the treacherous waters as he swims towards the nearby island. From this moment on, the suspense continues to rise as Rainsford meets the general and realizes that he will be hunted throughout Ship-Trap Island.

At the beginning of the story, Whitney, a hunting companion of Rainsford's, tells Rainsford:

The old charts call it "Ship-Trap Island." . . . A suggestive name, isn't it. Sailors have a curious dread of the place. I don't know why. Some superstition—

This quote foreshadows the horrors and extreme complications that Rainsford is going to experience later in the story. Later that night, as Rainsford is relaxing, he hears a strange "abrupt sound." He realizes that it is a gun being fired. The noise interests him, and he "leap[s] upon the rail and balance[s] himself there, to get greater elevation."

While balanced on the rail, his pipe falls out of his mouth, and he tries to catch it. In the process, he loses his balance and falls into the Caribbean Sea. The yacht speeds away and leaves him in the sea. This is already a great conflict, but Rainsford's troubles have only begun.

He is able to swim to the island. Ironically, he thinks that he is safe:

All he knew was that he was safe from his enemy, the sea, and that utter weariness was on him.

Soon after, Rainsford meets General Zaroff, a man who reveals that he hunts humans for sport:

I wanted the ideal animal to hunt . . . It must have courage, cunning, and above all, it must be able to reason.

Rainsford questions Zaroff, reminding him that "what [he] speak[s] of is murder." Zaroff, however, tells Rainsford that he has "romantic ideas about the value of human life." He does not think it is wrong to hunt humans.

Finally, Rainsford asks what happens if a man refuses to play General Zaroff's game. He explains,

I give him his option, of course. He need not play that game if he doesn't wish to. If he does not wish to hunt, I turn him over to Ivan. Ivan once had the honor of serving as official knouter to the Great White Czar, and he has his own ideas of sport.

Rainsford has no choice: he must try to evade General Zaroff, or he will be tortured by Ivan. Rainsford's yacht trip ends up being filled with weighty complications.

As General Zaroff explains to Rainsford, he has gotten bored with conventional hunting. Even big game animals no longer interest him because they no longer pose a challenge. Their instincts cannot compete against human reason. Therefore, the General has devised a new game: hunting humans.

As he explains to the horrified Rainsford, the rules are as follows: The General gives his human prey an "excellent" hunting knife and a supply of food. He then gives him a head start of three hours. After that time has elapsed, Zaroff heads after his quarry, armed with a pistol that can only shoot a short range. The hunt lasts for three days. If the hunted individual can elude Zaroff for three days, he wins. Otherwise, Zaroff wins by killing the human.

While Rainsford calls this kind of hunting murder, Zaroff believes he is giving his victims a sporting chance. However, he also notes that he has never lost such a hunt. He mentions as well that he has dogs, seven of which he lets out at night to track down anyone who tries to escape playing the game.

Rainsford becomes General Zaroff's prey. He outwits the General by pretending to jump into the ocean and commit suicide. However, he comes back later, confronts Zaroff, and kills him.

In any story that demonstrates the the classic organization of plot, an event has to occur (called the inciting incident) that sets the story in motion. If there isn't an initial problem, then there is nothing to complicate or to try to solve.

Had Rainsford not fallen off the boat, this would not have been possible. His departure from the boat is a turning point in the action of the story that helps also point to an eventual climax. The climax has to demonstrate a significant turning point as well. Usually that turning point is an answer to the inciting incident. If Rainsford hadn't fallen off the boat, he wouldn't have had to find an island, climb up to a place where someone lives, learn about that person, complicate a relationship with that person, and survive that relationship.

In my opinion, the event that starts the rising action is when Rainsford falls off the boat into the ocean.  I think that the rising action pretty much begins when he gets to Zaroff's place.

If that is the rising action, then it is started by the fact that Rainsford is on the island in the first place.  And why is he on the island?  He is on the island because he fell off the boat.

So that is why I think that him falling off the boat is the event that starts the rising action.

The event that sparks the main conflict in "The Most Dangerous Game" and sets the rising action in motion is Rainsford's realization that General Zaroff means to hunt HIM that night, and that he has no choice in the matter.  If he refuses to fight, then Zaroff will send Ivan out after him.  Either way, he will face one of them in a fight to the death.

Rainsford first learned of Zaroff's "hobby" during their conversation the night before.  Zaroff is very excited when he learns the identity of his island's newest arrival.  He tells Rainsford he has read his books on hunting before, and both men shares their stories and enthusiasm for hunting.

This conversation takes a turn, though, when Zaroff lets it be known that he has grown bored with most forms of hunting.  As he explains:

"Hunting had ceased to be what you call `a sporting proposition.' It had become too easy. I always got my quarry. Always. There is no greater bore than perfection.... No animal had a chance with me any more. That is no boast; it is a mathematical certainty. The animal had nothing but his legs and his instinct. Instinct is no match for reason. When I thought of this it was a tragic moment for me, I can tell you."

To combat this problem, Zaroff explains, he "had to invent a new animal to hunt," one with "the attributes of an ideal quarry."  These attributes, he continued, include "courage, cunning, and, above all, it must be able to reason."'

When Rainsford objects that no animal can reason, Zaroff insists that there is ONE who can.  Rainsford now understands Zaroff's "game," but says this is not hunting, but murder.  Rather than feeling chastised, Zaroff is amused by his words:

The general laughed with entire good nature. He regarded Rainsford quizzically. "I refuse to believe that so modern and civilized a young man as you seem to be harbors romantic ideas about the value of human life. Surely your experiences in the war--"

Rainsford does not let him finish his sentence, but rather makes it clear his wartime experiences did not make him "condone cold-blooded murder."  Still, Zaroff goes on laughing and says:

"I'll wager you'll forget your notions when you go hunting with me. You've a genuine new thrill in store for you, Mr. Rainsford."

The next day, Rainsford learns they are going hunting together, and he will be the prey.  The game is on - and the rising action takes off at a fast pace.

Suspense and conflict are immediately established in the opening of "The Most Dangerous Game."  The reader is immersed into a dialogue without benefit of description of character or setting.  Whitney, whose word choice creates suspense, tells Rainsford about a mysterious island that has the crew "a bit jumpy...even the tough-minded old Swede."  The night is dark without a breeze.  W. and R. disagree that evil is tangible and sailors have a sense for danger. 

That night R. sees nothing, but hears gunfire.  He strains to see, mystified. His pipe falls, he grabs for it, and falls into the water.  He hears screaming. (suspense)

R. wakes on shore after swimming for hours, sees a "mirage" of a chateau, meets Igor, and an enigmatic Gen. Z.  Their dinner conversation contains a conflict of opinion about what to hunt.  Suspense is created as the reader wonders about the "training school" and the hunt between R. and Z.; R. is given a choice:  hunt or deal with Ivan.  (suspense--what will Ivan do to R.?)

There is more suspense and conflict as R. is hunted by Z who gives R. another chance when his first trap fails.  R. struggles with himself to keep his nerve ( inner conflict) and create traps to stop Igor and Z.  It seems as though Z. will capture him even after Igor is killed, but R. jumps into the sea.  Suspense continues as the reader wonders if R. dies; R. reappears--then the ending.