The final debate in the New York City mayor’s race devolved into a chaotic contest Tuesday night marked by name-calling, lecturing, personal remarks and even profanity as the long-shot Republican candidate, Curtis Sliwa, sought to knock Eric Adams, the Democratic nominee, off-kilter at every turn.Mr. Sliwa faces extraordinarily difficult odds against Mr. Adams, and for much of the campaign, Mr. Adams has cast himself as a mayor-in-waiting who is already preparing to govern the nation’s largest city, ignoring Mr. Sliwa’s efforts to coax him into confrontation.But on Tuesday, the candidates did clash at times, and Mr. Sliwa spent much of the debate hectoring and interrupting Mr. Adams, and occasionally jolting him out of the rise-above-it-all demeanor that he deployed during their first debate last week. Mr. Adams lashed Mr. Sliwa for faking crimes and even over his record on child support.“That is scurrilous,” Mr. Sliwa protested.The two candidates staked out starkly different positions on matters from vaccine mandates to congestion pricing to outdoor dining, while finding common ground on some education and public safety issues.Still, the personal and political divide between the nominees was repeatedly thrown into sharp relief for viewers who tuned in one week before Election Day.Here are five takeaways from the debate:Mr. Sliwa needed an election-altering moment. He didn’t get one.Given New York City’s overwhelmingly Democratic tilt, any Republican nominee would face a steep climb in a mayoral contest. But Mr. Sliwa, whom Mr. Adams has referred to as a “clown,” may face an especially hard challenge.He has admitted, as Mr. Adams noted repeatedly, to faking crimes for publicity when he was younger. He is perhaps as well-known these days for owning more than a dozen cats as he is for any sweeping vision for the city. And while Mr. Sliwa has tried to make public safety a signature issue that galvanizes voters, that effort is complicated by Mr. Adams’s background as a former police officer.Taken together, Mr. Sliwa needed something of a miracle to change the seeming trajectory of the race — and he did not appear to get one. He did seem to catch Mr. Adams off guard at times, opening the debate by forcefully questioning Mr. Adams about interactions with gang members, which sent Mr. Adams veering into attack mode himself.But if Mr. Sliwa sought to produce any damaging new information about Mr. Adams that would make many voters seriously reconsider their choices, it was not immediately clear what that would be, since he pushed many familiar lines of attack.And as the debate wore on, Mr. Adams returned to his posture of ignoring Mr. Sliwa, looking at the camera instead of at his opponent, skipping opportunities to question or engage Mr. Sliwa, and insisting that his focus was on the voters of New York City.The two men clashed over one of the city’s biggest crises: homelessness.Curtis Sliwa, the Republican candidate for mayor, criticized Mayor Bill de Blasio’s social services commissioner, while his Democratic challenger, Eric Adams, called for building more housing and converting empty hotel rooms to address the crisis.Pool photo by Eduardo MunozHomelessness is one of the most pressing issues that the next mayor will face.There were nearly 48,000 homeless people, including almost 15,000 children, sleeping in the city’s shelter system every night in August, according to the Coalition for the Homeless.The overall figure for August also included 18,357 single adults, close to a record.Asked how they would tackle the homeless issue, Mr. Sliwa skirted the question and instead attacked Mr. Adams and his relationship with Mayor Bill de Blasio.The single adult population in homeless shelters has increased 60 percent since Mr. de Blasio took office in 2014. The mayor has cited homelessness as one of the issues he has struggled with the most during his two terms.“We’ve been out in the streets tending to their needs, getting them food and clothing, these lost souls,” Mr. Sliwa said before quickly pivoting to criticizing Mr. de Blasio and his social services commissioner, whom Mr. Adams has praised.“I would like you, Eric Adams, to condemn your partner and your teammate Bill de Blasio,” Mr. Sliwa said.Mr. Adams ignored Mr. Sliwa’s remark, calling homelessness a “real issue” before laying out a more detailed proposal..css-1xzcza9{list-style-type:disc;padding-inline-start:1em;}.css-3btd0c{font-family:nyt-franklin,helvetica,arial,sans-serif;font-size:1rem;line-height:1.375rem;color:#333;margin-bottom:0.78125rem;}@media (min-width:740px){.css-3btd0c{font-size:1.0625rem;line-height:1.5rem;margin-bottom:0.9375rem;}}.css-3btd0c strong{font-weight:600;}.css-3btd0c em{font-style:italic;}.css-1kpebx{margin:0 auto;font-family:nyt-franklin,helvetica,arial,sans-serif;font-weight:700;font-size:1.125rem;line-height:1.3125rem;color:#121212;}#NYT_BELOW_MAIN_CONTENT_REGION .css-1kpebx{font-family:nyt-cheltenham,georgia,’times new roman’,times,serif;font-weight:700;font-size:1.375rem;line-height:1.625rem;}@media (min-width:740px){#NYT_BELOW_MAIN_CONTENT_REGION .css-1kpebx{font-size:1.6875rem;line-height:1.875rem;}}@media (min-width:740px){.css-1kpebx{font-size:1.25rem;line-height:1.4375rem;}}.css-1gtxqqv{margin-bottom:0;}.css-16ed7iq{width:100%;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;-webkit-box-pack:center;-webkit-justify-content:center;-ms-flex-pack:center;justify-content:center;padding:10px 0;background-color:white;}.css-pmm6ed{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;}.css-pmm6ed > :not(:first-child){margin-left:5px;}.css-5gimkt{font-family:nyt-franklin,helvetica,arial,sans-serif;font-size:0.8125rem;font-weight:700;-webkit-letter-spacing:0.03em;-moz-letter-spacing:0.03em;-ms-letter-spacing:0.03em;letter-spacing:0.03em;text-transform:uppercase;color:#333;}.css-5gimkt:after{content:’Collapse’;}.css-rdoyk0{-webkit-transition:all 0.5s ease;transition:all 0.5s ease;-webkit-transform:rotate(180deg);-ms-transform:rotate(180deg);transform:rotate(180deg);}.css-eb027h{max-height:5000px;-webkit-transition:max-height 0.5s ease;transition:max-height 0.5s ease;}.css-6mllg9{-webkit-transition:all 0.5s ease;transition:all 0.5s ease;position:relative;opacity:0;}.css-6mllg9:before{content:”;background-image:linear-gradient(180deg,transparent,#ffffff);background-image:-webkit-linear-gradient(270deg,rgba(255,255,255,0),#ffffff);height:80px;width:100%;position:absolute;bottom:0px;pointer-events:none;}.css-1g3vlj0{font-family:nyt-franklin,helvetica,arial,sans-serif;font-size:1rem;line-height:1.375rem;color:#333;margin-bottom:0.78125rem;}@media (min-width:740px){.css-1g3vlj0{font-size:1.0625rem;line-height:1.5rem;margin-bottom:0.9375rem;}}.css-1g3vlj0 strong{font-weight:600;}.css-1g3vlj0 em{font-style:italic;}.css-1g3vlj0{margin-bottom:0;margin-top:0.25rem;}.css-19zsuqr{display:block;margin-bottom:0.9375rem;}.css-12vbvwq{background-color:white;border:1px solid #e2e2e2;width:calc(100% – 40px);max-width:600px;margin:1.5rem auto 1.9rem;padding:15px;box-sizing:border-box;}@media (min-width:740px){.css-12vbvwq{padding:20px;width:100%;}}.css-12vbvwq:focus{outline:1px solid #e2e2e2;}#NYT_BELOW_MAIN_CONTENT_REGION .css-12vbvwq{border:none;padding:10px 0 0;border-top:2px solid #121212;}.css-12vbvwq[data-truncated] .css-rdoyk0{-webkit-transform:rotate(0deg);-ms-transform:rotate(0deg);transform:rotate(0deg);}.css-12vbvwq[data-truncated] .css-eb027h{max-height:300px;overflow:hidden;-webkit-transition:none;transition:none;}.css-12vbvwq[data-truncated] .css-5gimkt:after{content:’See more’;}.css-12vbvwq[data-truncated] .css-6mllg9{opacity:1;}.css-qjk116{margin:0 auto;overflow:hidden;}.css-qjk116 strong{font-weight:700;}.css-qjk116 em{font-style:italic;}.css-qjk116 a{color:#326891;-webkit-text-decoration:underline;text-decoration:underline;text-underline-offset:1px;-webkit-text-decoration-thickness:1px;text-decoration-thickness:1px;-webkit-text-decoration-color:#326891;text-decoration-color:#326891;}.css-qjk116 a:visited{color:#326891;-webkit-text-decoration-color:#326891;text-decoration-color:#326891;}.css-qjk116 a:hover{-webkit-text-decoration:none;text-decoration:none;}Mr. Adams talked about his plan to turn 25,000 underused hotels rooms in the boroughs outside of Manhattan into permanent single-room occupancy housing for the homeless. Many hotels outside the main tourist and business districts in Manhattan were “built to be shelters,” Mr. Adams said.“We have to get out of the shelter business and get into the business of getting people permanent housing,” he said.Mr. Adams also said he would increase housing subsidies for families at risk of losing their homes, use a state law to get homeless people who can’t take care of themselves off the street, and partner with the police and mental health professionals to move homeless people out of the subways.“These are our neighbors. These are our former residents that lived next to us,” Mr. Adams said. “There’s a level of compassion that comes with it.”The debate turned nasty quickly.Eric Adams chastised Curtis Sliwa, saying he was acting immaturely, and Mr. Sliwa accused Mr. Adams of actually living in New Jersey.Pool photo by Eduardo MunozMr. Sliwa suggested that Mr. Adams consorted with murderers. Mr. Adams noted that Mr. Sliwa had admitted to faking crimes.And the debate had barely begun.On substantive issues, this debate proved similar to last week’s contest. But tonally, it proved far nastier.After Mr. Adams argued that he would have engaged more energetically with union leaders on vaccine mandates, Mr. Sliwa suggested that Mr. Adams talk to his “friend and teammate” Mr. de Blasio, who will soon be leaving office.“You are acting like my son when he was 4 years old,” Mr. Adams shot back. “Show some discipline so we can get to all of these issues. You’re interrupting, you’re being disrespectful.”Mr. Sliwa countered that Mr. Adams should stop being a “robot” and show compassion for city workers who risk losing their salaries for failing to get vaccinated against the coronavirus.Soon enough, the conversation got even more personal.Mr. Sliwa accused Mr. Adams of actually living in New Jersey, an allusion to questions that have been raised about Mr. Adams’s residency, and he mocked Mr. Adams’s decision to blame his tax-filing errors on his purportedly homeless accountant.“You fake where you live, Eric Adams,” Mr. Sliwa said.Mr. Adams said that Mr. Sliwa was demonstrating “clown-like actions,” and then accused him of hiding money so he would not have to pay child support.“That is scurrilous that you would say that,” Mr. Sliwa said. “How dare you bring my family into this?”The tenor of the debate did not go unnoticed.“I assume you’re not going to send each other holiday cards come December,” said Bill Ritter, who moderated the debate.On some hot-button issues, the candidates agreed — a reminder that the next mayor will not come from the city’s left wing.Eric Adams, the Democratic nominee, and his Republican challenger, Curtis Sliwa, both advocated increased policing in New York City to combat a rise in crime.Pool photo by Eduardo MunozMr. Adams and Mr. Sliwa may disagree on many of the specifics, but both fundamentally believe in expanding the role of the police in promoting public safety.Mr. Adams, who has said he was a victim of police brutality and spent much of his police career advocating for changes from within the system, also described his plan for bringing back an overhauled plainclothes unit to target gangs, “target those who are using guns.” His proposal has discomfited some New Yorkers who want to see the power of the police scaled back.And Mr. Sliwa indicated, in his typical forceful language, that he wants to empower the police to the greatest extent possible.Issues of education — and the best way to make public schools more integrated and equitable — do not necessarily break down along neat ideological lines. Both Mr. Adams and Mr. Sliwa have expressed concerns over Mr. de Blasio’s decision to end the gifted and talented program for elementary school children. They have said, instead, that they want to expand the program, positions that they revisited on Tuesday night.A moment of levity over pets and diet.When asked near the end of their debate to say something nice about each other, Eric Adams admired Curtis Sliwa’s dedication to saving cats and Mr. Sliwa praised Mr. Adams for choosing not to eat animals.Eduardo Munoz/ReutersFor a brief moment, the candidates did not fight with each other. They communed over animals. More precisely, Mr. Sliwa praised Mr. Adams’s decision to forgo eating animals, while Mr. Adams praised Mr. Sliwa’s work in rescuing them.The moment of bonhomie did not happen without some prodding.Toward the end of the debate, Mr. Ritter asked the candidates to say something “nice” about their opponent.“I take my hat off to Curtis, what he is doing with cats,” said Mr. Adams, perhaps referring to Mr. Sliwa’s advocacy for no-kill shelters, or perhaps to the more than a dozen cats that share a 320-square-foot studio apartment with Mr. Sliwa and his wife. “I think we need to be humane to all living beings.”Mr. Sliwa was even more effusive in his praise for Mr. Adams’s decision to become a vegan.“His promotion of a vegan way of life to avoid serious medical issues has probably already helped dozens, maybe hundreds, maybe thousands of people,” Mr. Sliwa said. “As someone who has been in the hospital many, many times, I hope one day to be a vegan.”Right now, Mr. Sliwa added, he is “at the vegetarian stage.” More