The indep endene numb ers and the hromati numb ers of - PowerPoint PPT Presentation

The indep enden e numb ers and the hromati numb ers of random subgraphs Andrei Raigo ro dskii Mos o w Institute of Physi s and T e hnology Mos o w, Russia A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 1 / 10

Main question A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

Main question Erd� os�R � enyi random graph Let n ∈ N , p ∈ [0 , 1] . G ( n, p ) on n is obtained b y dra wing indep endently edges y p . verti es, ea h with p robabilit A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

Main question Erd� os�R � enyi random graph Let n ∈ N , p ∈ [0 , 1] . G ( n, p ) on n is obtained b y dra wing indep endently edges y p . verti es, ea h with p robabilit Theo rem Let p b e a onstant o r a fun tion tending to zero and b ounded from b elo w b y a value c 1 where c > 1 . Let d = n 1 − p , . Then w.h.p. n α ( G ( n, p )) ∼ 2 log d ( np ) , χ ( G ( n, p )) ∼ 2 log d ( np ) . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

Main question Erd� os�R � enyi random graph Let n ∈ N , p ∈ [0 , 1] . G ( n, p ) on n is obtained b y dra wing indep endently edges y p . verti es, ea h with p robabilit Theo rem Let p b e a onstant o r a fun tion tending to zero and b ounded from b elo w b y a value c 1 where c > 1 . Let d = n 1 − p , . Then w.h.p. n α ( G ( n, p )) ∼ 2 log d ( np ) , χ ( G ( n, p )) ∼ 2 log d ( np ) . A general random subgraph Let n ∈ N , p ∈ [0 , 1] , G n = ( V n , E n ) graphs. G n,p � an a rbitra ry sequen e of is from G n of G n y p . obtained b y k eeping indep endently edges , ea h with p robabilit A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

Main question Erd� os�R � enyi random graph Let n ∈ N , p ∈ [0 , 1] . G ( n, p ) on n is obtained b y dra wing indep endently edges y p . verti es, ea h with p robabilit Theo rem Let p b e a onstant o r a fun tion tending to zero and b ounded from b elo w b y a value c 1 where c > 1 . Let d = n 1 − p , . Then w.h.p. n α ( G ( n, p )) ∼ 2 log d ( np ) , χ ( G ( n, p )) ∼ 2 log d ( np ) . A general random subgraph Let n ∈ N , p ∈ [0 , 1] , G n = ( V n , E n ) graphs. G n,p � an a rbitra ry sequen e of is from G n of G n y p . obtained b y k eeping indep endently edges , ea h with p robabilit out α ( G n,p ) and χ ( G n,p ) What an b e said ab ? A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

A sp e ial ase Main de�nition Let r, s, n ∈ N , s < r < n , let G ( n, r, s ) = ( V, E ) and , where V = { x = ( x 1 , . . . , x n ) : x i ∈ { 0 , 1 } , x 1 + . . . + x n = r } , E = {{ x , y } : ( x , y ) = s } . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 3 / 10

A sp e ial ase Main de�nition Let r, s, n ∈ N , s < r < n , let G ( n, r, s ) = ( V, E ) and , where V = { x = ( x 1 , . . . , x n ) : x i ∈ { 0 , 1 } , x 1 + . . . + x n = r } , E = {{ x , y } : ( x , y ) = s } . Equivalent de�nition Let r, s, n ∈ N , s < r < n . Let [ n ] an n -element b e set, and let G ( n, r, s ) = ( V, E ) , where � [ n ] � V = , E = { A, B ∈ V : | A ∩ B | = s } . r A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 3 / 10

A sp e ial ase Main de�nition Let r, s, n ∈ N , s < r < n , let G ( n, r, s ) = ( V, E ) and , where V = { x = ( x 1 , . . . , x n ) : x i ∈ { 0 , 1 } , x 1 + . . . + x n = r } , E = {{ x , y } : ( x , y ) = s } . Equivalent de�nition Let r, s, n ∈ N , s < r < n . Let [ n ] an n -element b e set, and let G ( n, r, s ) = ( V, E ) , where � [ n ] � V = , E = { A, B ∈ V : | A ∩ B | = s } . r out α ( G p ( n, r, s )) and χ ( G p ( n, r, s )) Again, what an b e said ab ? A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 3 / 10

Some motivation studying G ( n, r, s ) Why ? A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

Some motivation studying G ( n, r, s ) Why ? Co ding theo ry (�Johnson's graphs�): A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

Some motivation studying G ( n, r, s ) Why ? er α ( G ( n, r, s )) Co ding theo ry (�Johnson's graphs�): the indep enden e numb stands fo r the maximum size of a o de with one fo rbidden distan e; A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

Some motivation studying G ( n, r, s ) Why ? er α ( G ( n, r, s )) Co ding theo ry (�Johnson's graphs�): the indep enden e numb stands fo r the maximum size of a o de with one fo rbidden distan e; the er ω ( G (4 k, 2 k, k )) lique numb is resp onsible fo r the existen e of an Hadama rd matrix; et . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

Some motivation studying G ( n, r, s ) Why ? er α ( G ( n, r, s )) Co ding theo ry (�Johnson's graphs�): the indep enden e numb stands fo r the maximum size of a o de with one fo rbidden distan e; the er ω ( G (4 k, 2 k, k )) lique numb is resp onsible fo r the existen e of an Hadama rd matrix; et . geometry : G ( n, r, s ) Combinato rial is a � distan e � graph, i.e., its edges a re � 2( r − s ) er χ ( G ( n, r, s )) of the same length . The hromati numb p rovides imp o rtant b ounds in the Nelson�Hadwiger p roblems of spa e olo ring as w ell as in the Bo rsuk p roblem of pa rtitioning sets in spa es into pa rts of smaller diameter. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

Some motivation studying G ( n, r, s ) Why ? er α ( G ( n, r, s )) Co ding theo ry (�Johnson's graphs�): the indep enden e numb stands fo r the maximum size of a o de with one fo rbidden distan e; the er ω ( G (4 k, 2 k, k )) lique numb is resp onsible fo r the existen e of an Hadama rd matrix; et . geometry : G ( n, r, s ) Combinato rial is a � distan e � graph, i.e., its edges a re � 2( r − s ) er χ ( G ( n, r, s )) of the same length . The hromati numb p rovides imp o rtant b ounds in the Nelson�Hadwiger p roblems of spa e olo ring as w ell as in the Bo rsuk p roblem of pa rtitioning sets in spa es into pa rts of smaller diameter. G ( n, r, 0) graph; G ( n, 1 , 0) is the lassi al Kneser is just a omplete graph. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

Some motivation studying G ( n, r, s ) Why ? er α ( G ( n, r, s )) Co ding theo ry (�Johnson's graphs�): the indep enden e numb stands fo r the maximum size of a o de with one fo rbidden distan e; the er ω ( G (4 k, 2 k, k )) lique numb is resp onsible fo r the existen e of an Hadama rd matrix; et . geometry : G ( n, r, s ) Combinato rial is a � distan e � graph, i.e., its edges a re � 2( r − s ) er χ ( G ( n, r, s )) of the same length . The hromati numb p rovides imp o rtant b ounds in the Nelson�Hadwiger p roblems of spa e olo ring as w ell as in the Bo rsuk p roblem of pa rtitioning sets in spa es into pa rts of smaller diameter. G ( n, r, 0) graph; G ( n, 1 , 0) is the lassi al Kneser is just a omplete graph. Constru tive b ounds fo r Ramsey numb ers. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

of G ( n, r, s ) : Random subgraphs indep enden e numb ers Theo rem (F rankl, F� uredi, 1985) Let r, s as n → ∞ . b e �xed then α ( G ( n, r, s )) = Θ ( n s ) . If r � 2 s + 1 , A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 5 / 10

of G ( n, r, s ) : Random subgraphs indep enden e numb ers Theo rem (F rankl, F� uredi, 1985) Let r, s as n → ∞ . b e �xed then α ( G ( n, r, s )) = Θ ( n s ) . If r � 2 s + 1 , � n r − s − 1 � If r > 2 s + 1 , then α ( G ( n, r, s )) = Θ . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 5 / 10

of G ( n, r, s ) : Random subgraphs indep enden e numb ers Theo rem (F rankl, F� uredi, 1985) Let r, s as n → ∞ . b e �xed then α ( G ( n, r, s )) = Θ ( n s ) . If r � 2 s + 1 , � n r − s − 1 � If r > 2 s + 1 , then α ( G ( n, r, s )) = Θ . Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 2013�2016) Let r, s as n → ∞ . b e �xed If r � 2 s + 1 , w.h.p. α ( G 1 / 2 ( n, r, s )) = Θ ( α ( G ( n, r, s )) log n ) then . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 5 / 10

of G ( n, r, s ) : Random subgraphs indep enden e numb ers Theo rem (F rankl, F� uredi, 1985) Let r, s as n → ∞ . b e �xed then α ( G ( n, r, s )) = Θ ( n s ) . If r � 2 s + 1 , � n r − s − 1 � If r > 2 s + 1 , then α ( G ( n, r, s )) = Θ . Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 2013�2016) Let r, s as n → ∞ . b e �xed If r � 2 s + 1 , w.h.p. α ( G 1 / 2 ( n, r, s )) = Θ ( α ( G ( n, r, s )) log n ) then . If r > 2 s + 1 , w.h.p. α ( G 1 / 2 ( n, r, s )) ∼ α ( G ( n, r, s )) then . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 5 / 10