Rainbow triangles in 3-edge-colored graphs J´ ozsef Balogh, Ping Hu, Bernard Lidick´ y, Florian Pfender, Jan Volec, Michael Young SIAM Conference on Discrete Mathematics Jun 17, 2014
Problem Find a 3-edge-coloring of a complete graph K n maximizing the number of copies of rainbow colored triangles . 2
Problem Find a 3-edge-coloring of a complete graph K n maximizing the number of copies of rainbow colored triangles . Color edges randomly, expected density 2 9 . 2
Problem Find a 3-edge-coloring of a complete graph K n maximizing the number of copies of rainbow colored triangles . Color edges randomly, expected density 2 9 . Iterated blow-up of triangle 1 4 = . denotes graph and/or its density 2
F ( n ) = max # of over all 3-edge-colorings of K n 3
F ( n ) = max # of over all 3-edge-colorings of K n Conjecture (Erd˝ os and S´ os; ’72 − ) For all n > 0 , F ( n ) = F ( a ) + F ( b ) + F ( c ) + F ( d ) + abc + abd + acd + bcd , where a + b + c + d = n; a , b , c , d are as equal as possible, and F (0) = 0 . a b c d 3
F ( n ) = max # of over all 3-edge-colorings of K n Conjecture (Erd˝ os and S´ os; ’72 − ) For all n > 0 , F ( n ) = F ( a ) + F ( b ) + F ( c ) + F ( d ) + abc + abd + acd + bcd , where a + b + c + d = n; a , b , c , d are as equal as possible, and F (0) = 0 . a b c d 3
F ( n ) = max # of over all 3-edge-colorings of K n Conjecture (Erd˝ os and S´ os; ’72 − ) For all n > 0 , F ( n ) = F ( a ) + F ( b ) + F ( c ) + F ( d ) + abc + abd + acd + bcd , where a + b + c + d = n; a , b , c , d are as equal as possible, and F (0) = 0 . a b c d 0 . 4 = 3
Flag algebras application Construction: 0 . 4 ≤ • get a matching upper bound ≈ 0 . 4 • round the result • get subgraphs with 0 density • get extremal construction (stability) 4
Flag algebras application Construction: 0 . 4 ≤ • get a matching upper bound ≈ 0 . 4 • round the result • get subgraphs with 0 density • get extremal construction (stability) Flag algebras (on 6 vertices) give only ≤ 0 . 4006 , not enough for rounding. 4
Flag algebras application Construction: 0 . 4 ≤ • get a matching upper bound ≈ 0 . 4 • round the result • get subgraphs with 0 density • get extremal construction (stability) Flag algebras (on 6 vertices) give only ≤ 0 . 4006 , not enough for rounding. The iterative extremal construction is causing troubles.... 4
Not iterated extremal constructions Theorem ( Tur´ an) n 5 n n # of edges over K l -free graphs is maximized by 5 5 n n 5 5 a , Theorem ( Hatami, Hladk´ y, Kr´ l, Norine, Razborov) n 5 # of C 5 s over triangle-free graphs is maximized n n 5 5 by n n 5 5 a , Theorem ( Cummings, Kr´ l, Pfender, Sperfeld, Treglown, Young) n 5 # of monochromatic triangles over n n 5 5 3-edge-colored graphs is minimized by n n 5 5 And more... http://flagmatic.org ( n large enough) 5
Iterated extremal constructions Theorem (Falgas-Ravry, Vaughan) # of and is maximized by Theorem (Huang) # of . is maximized by . . a , Theorem (Hladk´ y, Kr´ l, Norine) # of is maximized by 6
Our main result F ( n ) = max # of over all coloring of K n Theorem ( Balogh, Hu, L., Pfender, Volec, Young) For all n > n 0 , F ( n ) = F ( a ) + F ( b ) + F ( c ) + F ( d ) + abc + abd + acd + bcd , where a + b + c + d = n; a , b , c , d are as equal as possible. a b c d 7
Sketch of proof Goal: maximizing gives edge-coloring like 8
Sketch of proof Goal: maximizing gives edge-coloring like • pick a properly 3-edge-colored K 4 8
Sketch of proof Goal: maximizing gives edge-coloring like • pick a properly 3-edge-colored K 4 8
Sketch of proof Goal: maximizing gives edge-coloring like • pick a properly 3-edge-colored K 4 8
Sketch of proof Goal: maximizing gives edge-coloring like X 1 X 2 • pick a properly 3-edge-colored K 4 • partition the rest X 4 X 3 8
Sketch of proof Goal: maximizing gives edge-coloring like X 1 X 2 • pick a properly 3-edge-colored K 4 • partition the rest X 4 X 3 8
Sketch of proof Goal: maximizing gives edge-coloring like X 1 X 2 • pick a properly 3-edge-colored K 4 • partition the rest X 4 X 3 8
Sketch of proof Goal: maximizing gives edge-coloring like X 1 X 2 • pick a properly 3-edge-colored K 4 • partition the rest X 4 X 3 8
Sketch of proof Goal: maximizing gives edge-coloring like X 1 X 2 • pick a properly 3-edge-colored K 4 • partition the rest • correct edges between X i s X 4 X 3 8
Sketch of proof Goal: maximizing gives edge-coloring like X 1 X 2 • pick a properly 3-edge-colored K 4 • partition the rest • correct edges between X i s • no orange trash X 4 X 3 8
Sketch of proof Goal: maximizing gives edge-coloring like X 1 X 2 • pick a properly 3-edge-colored K 4 • partition the rest • correct edges between X i s • no orange trash • balance sizes of X i s X 4 X 3 8
Sketch of proof Goal: maximizing gives edge-coloring like • pick a properly 3-edge-colored K 4 • partition the rest • correct edges between X i s • no orange trash • balance sizes of X i s 8
Sketch of proof Goal: maximizing gives edge-coloring like X 1 X 2 • pick a properly 3-edge-colored K 4 • partition the rest • correct edges between X i s • no orange trash • balance sizes of X i s X 4 X 3 How to pick the properly 3-edge-colored K 4 ? ( | X i | s close to 0 . 25 n , few wrongly colored edges, small trash) 8
How to pick K 4 ? Use Flag Algebras! 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing ( n − 5) ≥ 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing 1 � n � ( n − 5) ≥ ( n − 5) � 4 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing � n � 2 1 � n 5 � ( n − 5) ≥ ( n − 5) = � n � � 4 4 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing � n � 2 2 1 � n 5 � ( n − 5) ≥ ( n − 5) = � = ( n − 5) � n � 5 4 4 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing � n � 2 2 1 � n 5 � ( n − 5) ≥ ( n − 5) = � = ( n − 5) � n � 5 4 4 FA: ≥ 0 . 4 then > 0 . 23516 , < 0 . 0952 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing > 0 . 988 � n � 2 2 1 � n 5 � ( n − 5) ≥ ( n − 5) = � = ( n − 5) � n � 5 4 4 FA: ≥ 0 . 4 then > 0 . 23516 , < 0 . 0952 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing > 0 . 988 � n � 2 2 1 � n 5 � ( n − 5) ≥ ( n − 5) = � = ( n − 5) � n � 5 4 4 FA: ≥ 0 . 4 then > 0 . 23516 , < 0 . 0952 Result for K n : | X 1 | + | X 2 | + | X 3 | + | X 4 | > 0 . 988( n − 5) 9
How to pick K 4 ? Use Flag Algebras! Try 1: Pick maximizing > 0 . 988 � n � 2 2 1 � n 5 � ( n − 5) ≥ ( n − 5) = � = ( n − 5) � n � 5 4 4 FA: ≥ 0 . 4 then > 0 . 23516 , < 0 . 0952 Result for K n : | X 1 | + | X 2 | + | X 3 | + | X 4 | > 0 . 988( n − 5) Balancing needed... 9
How to pick K 4 ? Use Flag Algebras! Try 2: Pick maximizing − 26 + + 9 10
How to pick K 4 ? Use Flag Algebras! Try 2: Pick maximizing − 26 + + 9 FA: 4 − 26 + + > 0 . 002629 15 45 10
How to pick K 4 ? Use Flag Algebras! Try 2: Pick maximizing − 26 + + > 0 . 0276 9 FA: 4 − 26 + + > 0 . 002629 15 45 10
How to pick K 4 ? Use Flag Algebras! Try 2: Pick maximizing − 26 + + > 0 . 0276 9 FA: 4 − 26 + + > 0 . 002629 15 45 Final equation: � � | X i | 2 > 0 . 0276 n 2 | X i || X j | − | F | − 26 2 9 1 ≤ i < j ≤ 4 1 ≤ i ≤ 4 F = wrongly colored edges. 10
How the first step worked � � | X i | 2 > 0 . 0276 n 2 | X i || X j | − | F | − 26 2 9 1 ≤ i < j ≤ 4 1 ≤ i ≤ 4 Implies: X 1 X 2 0 . 244 n < | X i | < 0 . 256 n | Trash | < 0 . 006 n � n � | F | < 0 . 00008 2 X 4 X 3 F = wrongly colored edges. 11
More results Theorem # of rainbow K 3 s is maximized by if on 4 k vertices. 12
More results Theorem Theorem # of rainbow K 3 s is # of induced C 5 s is maximized by maximized by if on 4 k vertices. if on 5 k vertices. 12
More results Theorem Theorem Theorem # of rainbow K 3 s is # of induced C 5 s is # of induced oriented maximized by maximized by C 4 s is maximized by if on 4 k vertices. if on 5 k vertices. if on 4 k vertices. 12
More results Theorem Theorem Theorem # of rainbow K 3 s is # of induced C 5 s is # of induced oriented maximized by maximized by C 4 s is maximized by if on 4 k vertices. if on 5 k vertices. if on 4 k vertices. (for all k ) 12
Thank you for listening! 13
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