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Drawings of dense graphs have many crossings
of graph drawing, the crossing number inequality or crossing lemma gives a lower bound on the minimum number of edge crossings in a plane drawing of a
Crossing_number_inequality
Fewest edge crossings in drawing of a graph
crossing numbers of complete bipartite graphs remains unproven, as does an analogous formula for the complete graphs. The crossing number inequality states
Crossing number (graph theory)
Crossing_number_(graph_theory)
Bound on the number of incidences between points and lines in the plane
Later, László Székely discovered a much simpler proof using the crossing number inequality for graphs. This method has been used to produce the explicit
Szemerédi–Trotter_theorem
2018 mathematics book by Marcus Schaefer
the Zarankiewicz crossing number conjecture), again giving a conjectured formula). It also includes the crossing number inequality, and the Hanani–Tutte
Crossing_Numbers_of_Graphs
Subdivision of the plane by lines
O(n^{2})} crossings in this drawing. However, by the crossing number inequality, there are Ω ( x 3 / m 2 ) {\displaystyle \Omega (x^{3}/m^{2})} crossings. In
Arrangement_of_lines
Geometric graph with unit edge lengths
between points and unit circles, and is closely related to the crossing number inequality and to the Szemerédi–Trotter theorem on incidences between points
Unit_distance_graph
Geometry book
problem to other topics including the Cauchy–Schwarz inequality, the crossing number inequality, the Szemerédi–Trotter theorem on incidences between points
The_Erdős_Distance_Problem
Hungarian-American computer scientist
proved the crossing number inequality, that any drawing of a graph with n vertices and m edges, where m > 4n, has at least m3 / 100n2 crossings. Ajtai and
Miklós_Ajtai
Relation between graph coloring and crossings
otherwise greedy coloring would use fewer colors) together with the crossing number inequality according to which every graph G = ( V , E ) {\displaystyle G=(V
Albertson_conjecture
Czech-Canadian mathematician
connectivity and maximum independent set size of a graph, earned Chvátal his Erdős number of 1. Specifically, if there exists an s such that a given graph is s-vertex-connected
Václav_Chvátal
Points separated from others by a line
{\displaystyle O(nk^{1/3})} , as was shown by Tamal Dey using the crossing number inequality of Ajtai, Chvátal, Newborn, and Szemerédi. However, the best known
K-set_(geometry)
Hungarian-American mathematician
proved the famous crossing number inequality, that a graph with n vertices and m edges, where m > 4n has at least m3 / 64n2 crossings. With Paul Erdős
Endre_Szemerédi
probabilistic proof. Crossing number inequality which is a lower bound on the number of crossing for any drawing of a graph as a function of the number of vertices
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
Correlation inequality
In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic
FKG_inequality
Ways inequality is measured
Income inequality metrics or income distribution metrics are used by social scientists to measure the distribution of income and economic inequality among
Income_inequality_metrics
economy of the People's Republic of China has a high degree of income inequality. According to the Asian Development Bank Institute, "before China implemented
Income_inequality_in_China
Large number used in number theory
on its value. The exact value of Skewes's number is still not known, but it is known that there is a crossing between π ( x ) < li ( x ) {\displaystyle
Skewes's_number
Hungarian mathematician
This situation led Turan to consider how to achieve the minimum number of crossings for m kilns and n warehouses. It was only after the war, in 1952
Pál_Turán
Gender Inequality Index (GII). Among the GII components, China's maternal mortality ratio was 23 out of 100,000 live births. In its education inequality, 68
Gender_inequality_in_China
Number, approximately 3.14
The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants. Wirtinger's inequality also generalizes
Pi
Smallest number of edges of an equivalent polygonal path for a knot
These inequalities are both tight for the trefoil knot, which has a crossing number of 3 and a stick number of 6. The upper bound on the stick number does
Stick_number
Existence of a line through two points
total number of edges would be at least 3 V {\displaystyle 3V} , contradicting this inequality. Therefore, some vertices must be the crossing point of
Sylvester–Gallai_theorem
On least area of curves of constant width
possible to bound the number of missed shots by O ( log log n ) {\displaystyle O(\log \log n)} . By the isoperimetric inequality, the curve of constant
Blaschke–Lebesgue_theorem
Ongoing international incident
was almost 40% in 2025. Morocco has high levels of inequality, and between 2019 and 2022, inequality and poverty increased. In September and October 2025
2026_Ceuta_migrant_crisis
Prime Minister of the United Kingdom since 2026
"business-friendly socialism" and "a modern and functional response to the high-inequality, low-growth trap that came from the 1980s drive to privatise economic
Andy_Burnham
Mathematical theory of knots
switching all crossings, one can convert a grid diagram into the associated Legendrian knot. In his thesis , Daniel Bennequin proved an inequality involving
Thurston–Bennequin_number
Country in East Asia
agricultural economy and rapidly urbanized, although this led to stark economic inequality between urban and rural areas, as well as environmental deterioration
China
Waterway entrance in Victoria, Australia
outside this flat to 30m and inside the Heads to as much as 90 m. This inequality of depth combined with tidal streams running up to 6 knots, causes the
The_Rip
Natural number, composite number
making it the first even nontotient. According to the Shapiro inequality, 14 is the least number n {\displaystyle n} such that there exist x 1 {\displaystyle
14_(number)
Economic anthropologist (born 1982)
Royal Society of Arts, a visiting senior fellow at the International Inequalities Institute at the London School of Economics, and was the Chair of Global
Jason_Hickel
International border in North America
Churubusco–Franklin Centre Border Crossing, where travelers may enter only the United States; and the Four Falls Border Crossing, where travelers may enter only
Canada–United_States_border
announced a range of wealth, inheritance and corporate taxes to "tackle inequality and corporate greed." They also proposed a tax-free threshold for incomes
2026 New Zealand general election
2026_New_Zealand_general_election
Compact astronomical body
boundary of no escape is called the event horizon. In general relativity, crossing a black hole's event horizon traps an object inside but produces no locally
Black_hole
and 366,872 kilometers of other roads. The network includes a significant number of high-capacity roads, with 2,160 kilometers of motorway and 940 kilometers
Roads_in_South_Africa
Mathematical puzzle of avoiding crossings
water, gas and electricity, is a mathematical puzzle that asks for non-crossing connections to be drawn between three houses and three utility companies
Three_utilities_problem
Flag used by the transgender community
community. Transgender flags at the Prague Pride parade. The parade is crossing the Čechův Bridge, 2015. Transgender flag at the Baltimore Pride parade
Transgender_flag
Special administrative region of China
amount of income inequality, which has risen since the handover, as the region's ageing population has gradually added to the number of nonworking people
Hong_Kong
Self-identification collected by the US census
and Ethnic Data on the U.S. Census and the Implications for Tracking Inequality," Sociology of Race and Ethnicity 4, no. 1 (2018): 1–13. Reynolds, Gerald
Race and ethnicity in the United States census
Race_and_ethnicity_in_the_United_States_census
non-singular algebraic variety over algebraic number field that not satisfy certain height inequality are contained in some Zariski-closed set. n conjecture:
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
2024 civilian protests during Gaza humanitarian crisis
of humanitarian aid to the Gaza Strip at the Kerem Shalom crossing, at the Nitzana crossing, at the Port of Ashdod, and on the roads that aid trucks traveled
Israeli direct action against aid delivery to Gaza
Israeli_direct_action_against_aid_delivery_to_Gaza
2011 American protest movement
) was a left-wing populist and progressive movement against economic inequality, capitalism, corporate greed, big finance, and the influence of money
Occupy_Wall_Street
Country in West Asia
Underwater Electricity Cable". Arutz Sheva. Retrieved 7 January 2017. "Inequality Report: The Palestinian Arab Minority in Israel – Adalah". www.adalah
Israel
NP-hard problem in combinatorial optimization
with crossings can be made into a shorter solution without crossings by local optimizations. The Euclidean distance obeys the triangle inequality, so the
Travelling_salesman_problem
Limiting case which is different from the rest of the class
degenerate triangle is a triangle that does not satisfy the strict triangle inequality: a + b > c {\displaystyle a+b>c} , instead, it satisfies: a + b = c {\displaystyle
Degeneracy_(mathematics)
people smugglers are directing migrants to pay for English Channel crossings using a number of UK-registered companies. Channel 4 removes all episodes of Married
2026_in_the_United_Kingdom
Quantified formulas with real-number variables
real number values, and where F ( X 1 , … X n ) {\displaystyle F(X_{1},\dots X_{n})} is a quantifier-free formula involving equalities and inequalities of
Existential theory of the reals
Existential_theory_of_the_reals
Number, approximately 1.618
so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations, which states that for every irrational
Golden_ratio
Country in West Africa
The 100%-state-owned filling station company, Ghana Oil Company, is the number one petroleum and gas filling station, and the 100%-state-owned state oil
Ghana
2024 film festival in Berlin, Germany
the occupied West Bank. This situation of apartheid between us, this inequality, has to end". Berlin Mayor, Kai Wegner, and numerous other German politicians
74th Berlin International Film Festival
74th_Berlin_International_Film_Festival
Prime Minister of Israel (1996–1999; 2009–2021; 2022–present)
concentrated economy to competition to lower consumer prices, reduce income inequality, and increase economic growth. Netanyahu had formed the Concentration
Benjamin_Netanyahu
Polyhedron with parallel bases connected by triangles
excluded from counting. The number of uniform crossed antiprisms for fixed p can be determined using simple inequalities. The condition on possible q
Antiprism
2020 video game
passing since the original game was released, the themes of economic inequality, corporate monopoly, and environmentalism were still relevant to the current
Final_Fantasy_VII_Remake
Pattern in storytelling
elixir. They're navigating our patriarchal society with unequal pay and inequalities. In the final chapter, they may end up on equal footing. But when you
Hero's_journey
Luxembourgish research institute
within the EU, political instability and global inequality, more and more people have been crossing borders in recent decades, either as refugees, regular
Luxembourg Institute of Socio-Economic Research
Luxembourg_Institute_of_Socio-Economic_Research
Mainland period of the Republic of China
both the central and northern provinces of China, and short-lived. The number of acts passed by the government was few and included the formal abdication
Republic_of_China_(1912–1949)
Second election of Donald Trump
feels more tangible than blaming systemic problems like rising economic inequality and the difficulty of adapting to technological and cultural changes."
2024 United States presidential election
2024_United_States_presidential_election
modernization of the tax system and labor laws, and the reduction of income inequality. Tax revenues, at 19.6 percent of GDP in 2013, were the lowest among the
Economy_of_Mexico
1849–1850 novel by Charles Dickens
clearly attacks the official status of marriage, which perpetuated an inequality between the sexes, an injustice that does not end with the separation
David_Copperfield
Military reserve force of the Chinese People's Liberation Army (PLA)
full service. The reserve forces are organized as full units with a small number of active servicemen as an organizational skeleton, plus a larger body of
People's Liberation Army Reserve Force
People's_Liberation_Army_Reserve_Force
Political instability c. 134–30 BC
the breakdown of the political code known as the mos maiorum, widening inequality among wealth and land, and class conflict between the traditional aristocracy
Crisis_of_the_Roman_Republic
Political slang
time, woke came to be used to refer to a broader awareness of social inequalities such as sexism and denial of LGBTQ rights. Woke has also been used as
Woke
Country in West Asia
needed] The Rafah Crossing is the only land crossing between the Gaza Strip and Egypt. There exists also the Kerem Shalom border crossing, managed by Israel
Palestine
they will "accelerate land reform and redistribution to reduce asset inequality and protect security of tenure, improve food security and agricultural
2024 South African general election
2024_South_African_general_election
Major river in the western United States and Mexico
remote Pahreah Crossing in Arizona, where he took over a ferry first established in 1864 by Jacob Hamblin. This, the only river crossing for hundreds of
Colorado_River
Archipelagic country in Southeast Asia
region. Despite its fast economic growth, it continues to struggle with inequality, widespread corruption, and vulnerability to natural disasters due to
Philippines
Energy level splitting due to strong light-matter coupling
avoided crossing in dispersion relations. When the frequencies of the uncoupled photon and matter modes are tuned into resonance, instead of crossing, the
Rabi_splitting
Wave breaking steepness limit
non-breaking waves, and an upper theoretical limit for wave breaking. If the inequality is violated at a point, a steady periodic wave cannot exist and breaking
Miche_criterion
opening up Special economic zones E-commerce Online advertising Economic inequality in China Energy Coal Petroleum Renewable Forced evictions Foreign aid
Public_holidays_in_China
Mathematical problem involving optimal stopping theory
X_{n}} , such that for any stopping rule τ {\displaystyle \tau } , the inequality P r ( X τ = max i ∈ 1 : n X i ) ≤ max r ∈ 1 : n r − 1 n ∑ i = r n 1 i
Secretary_problem
Country in Oceania
original on 2 June 2020. Retrieved 15 May 2020. "Poverty – Poverty and Inequality". "Small island economies" (PDF). Asian Development Bank. 2013. Archived
Australia
Zoran Milanović, ran on an anti-corruption campaign, opposing social inequality and growing inflation, which he attributed to policies of the HDZ government
2024–25 Croatian presidential election
2024–25_Croatian_presidential_election
International border in North America
in the world, with approximately 350 million documented crossings annually. Illegal crossing of the border to enter the United States has caused the Mexico–United
Mexico–United_States_border
Country in East Asia
Share of Disposable Income by Quintile Group of Households and Income Inequality Indices". Statistical Bureau. Archived from the original on 8 September
Taiwan
Island country in Southeast Asia
US$3 billion in revenue. In 2025, Singapore saw a record number of nearly 245 million people crossing its borders. In the same year, Singapore turned away
Singapore
Israeli-American mathematician (born 1944)
integral inequalities", Bull. Amer. Math. Soc., vol. 80, 1974, pp. 715–717. doi:10.1090/S0002-9904-1974-13565-2 with Samuel Karlin: "Some inequalities for
Shmuel_Friedland
Stability conditions for triangulated cateogires
{rank} (-)} is the slope function. This inequality of slopes can be equivalently stated by saying the complex number Z ( W ) = − deg ( W ) + i rank (
Bridgeland stability condition
Bridgeland_stability_condition
Country in West Africa
poverty rates in producing municipalities. Nevertheless, it also increased inequality and child labor. The value of Burkina Faso's exports fell from $2.77 billion
Burkina_Faso
Cohort born from 1946 to 1964
dim prospects despite having attained a high level of education. Income inequality, stagnating or declining real wages, and growing public debt were contributory
Baby_boomers
British politician (born 1982)
2018). "The UK Treasury's Robert Jenrick talks to Business Insider about inequality, productivity and GDP growth in the UK". Business Insider. Retrieved 10
Robert_Jenrick
authority or trust over the victim. There is no close-in-age exception, crossing the age boundary is Criminal Sexual Assault. Any sexual contact of minors
Age of consent in the United States
Age_of_consent_in_the_United_States
2021 film by Chen Kaige, Tsui Hark and Dante Lam
intended to control the grievances caused by rising unemployment and inequality". She further criticized China's interpretation of the Korean War, as
The_Battle_at_Lake_Changjin
President of El Salvador since 2019
by Mexican immigrants. Much of the gang violence stemmed from income inequality, poverty, poor schools, a lack of job opportunities, and high urbanization
Nayib_Bukele
industrialization were imperative for a competitive China. In response, a number of scholars resisted the idea, saying democracy and Westernization had no
Democracy_in_China
in Cross-Harbour Tunnel and Eastern Harbour Crossing from its previous price a drop would induce inequality to their public users. DAB Legislator Ben Chan
Congestion pricing in Hong Kong
Congestion_pricing_in_Hong_Kong
Graphs whose distances obey Ptolemy's inequality
graph is an undirected graph whose shortest path distances obey Ptolemy's inequality, which in turn was named after the Greek astronomer and mathematician
Ptolemaic_graph
has formally recognized 139 residential schools across Canada, but this number excludes schools that operated without federal support. The last school
List of Indian residential schools in Canada
List_of_Indian_residential_schools_in_Canada
Gender identity of lacking a gender
Early American Studies (University of Pennsylvania Press), volume 12, number 3, Fall 2014, pp. 576–600 Miller, sj. "About Dr. sj Miller". sjmiller.info
Agender
Capital city of the United States
Retrieved December 26, 2017. "Washington DC Bridges: A Guide to River Crossings". Trip Savvy. Retrieved September 17, 2022. Clabaugh, Jeff (October 2
Washington,_D.C.
1989 crime in New York City
case became a prominent example of racial profiling, discrimination, and inequality in the legal system and the media. All five defendants sued the City of
Central_Park_jogger_case
Cohort born from 1981 to 1996
exacerbating household income inequality". The Economist. 21 September 2017. Retrieved 20 December 2020. "Sex, brains and inequality". The Economist. 8 February
Millennials
Children's board game
"Jennifer O'Connell, Mom, And 6-Year-Old Daughter Ask Hasbro About Gender Inequality In 'Guess Who?'". The Huffington Post. 21 November 2012. "Guess Who? -
Guess_Who?
Political ideologies favouring social orders
property, religion, or tradition.[excessive citations] Hierarchy and inequality may be seen as natural results of traditional social differences or competition
Right-wing_politics
American politician and activist (born 1989)
"authoritarian", criticized activist detentions, drew attention to economic inequality, and called for the "courage to brawl for the working class". Ocasio-Cortez
Alexandria_Ocasio-Cortez
Shape with three equal sides
arcs on its sides, a result of the Reuleaux triangle Many theorems, inequalities, and problems involves equilateral triangles. For example, Napoleon's
Equilateral_triangle
partly invested in a Future Fund to reduce the national debt. Income inequality and private debt increased as the economy expanded, with the biggest increase
History_of_Australia
Four-sided polygon
interior angle sum formula: S = (n − 2) × 180° (here, n=4). All non-self-crossing quadrilaterals tile the plane, by repeated rotation around the midpoints
Quadrilateral
American World War II wartime poster
"We Women", uniting all women in a sisterhood fighting against gender inequality. This was very different from the poster's 1943 use to control employees
We_Can_Do_It!
Symbol of communism
League for the Fifth International merges a hammer with the number 5, using the number's lower arch to form the sickle. A sickle with a rifle is also
Hammer_and_sickle
Political alignment in the right-wing spectrum
inequalities between people to be artificial and negative, which should be overcome by an active state, whereas the right believes that inequalities between
Far-right_politics
Argentine revolutionary (1928–1967)
attempted to destroy any government that sought to redress the socioeconomic inequality endemic to Latin America and other developing countries. Guevara stated:
Che_Guevara
travel, tourism, insurance
CROSSING NUMBER-INEQUALITY
CROSSING NUMBER-INEQUALITY
Boy/Male
Christian & English(British/American/Australian)
At the Crossing
Girl/Female
Muslim American Arabic English Gaelic
Jewel. Amber stone.
Male
English
English form of Norman Germanic Huncberct, possibly HUMBERT means "bright support."Â
Girl/Female
English American
Born during the summer.
Surname or Lastname
English
English : habitational name from any of the various places so called from their situation on a stream with this name. Humber is a common prehistoric river name, of uncertain origin and meaning.
Boy/Male
American, Anglo, Australian, British, Chinese, Christian, English
Wide River-crossing
Surname or Lastname
English
English : variant of Sumpter.Fort Sumter, SC, was named in honor of Thomas Sumter, known as the ‘Gamecock of the Revolution’ for the fear he inspired in the British and Tory forces and the pivotal role he played in key American victories. Born in 1734 near Charlottesville, VA, he was of Welsh heritage; his ancestors probably emigrated to America in the late 17th century.
Female
English
English name derived from the vocabulary word, summer, from Old English sumor, SUMMER means "summer," the hot season of the year.
Female
Native American
Native American Algonquin name NUMEES means "sister."
Boy/Male
Indian, Sanskrit
Crossing Shiva
Surname or Lastname
English
English : perhaps a variant of Crossland.
Girl/Female
Hindu, Indian
Number; Definition
Male
German
German byname BAMBER means "short and fat."Â
Girl/Female
American, Arabic, Australian, British, Chinese, English, Hebrew
The Warmest Season of the Year; Summer Season; Name of the Season; Summer; The Hot Season of the Year
Boy/Male
Hindu, Indian
Number
Boy/Male
English American Shakespearean
River crossing.
Surname or Lastname
English
English : occupational name for a summoner, an official who was responsible for ensuring the appearance of witnesses in court, Middle English sumner, sumnor.William Sumner came to Dorchester, MA, from England in about 1635. His descendants include U.S. Senator Charles Sumner, a major force in the struggle to end slavery, who was born in 1811 in Boston.
Boy/Male
Tamil
The number
Surname or Lastname
English
English : perhaps a variant of Pamber, a habitational name from a place in Hampshire named Pamber, from Old English penn ‘fold’, ‘enclosure’ + beorg ‘hill’.
Boy/Male
Hindu
The number
CROSSING NUMBER-INEQUALITY
CROSSING NUMBER-INEQUALITY
CROSSING NUMBER-INEQUALITY
CROSSING NUMBER-INEQUALITY
CROSSING NUMBER-INEQUALITY
CROSSING NUMBER-INEQUALITY
CROSSING NUMBER-INEQUALITY
travel, tourism, insurance