LINEAR ALGEBRA METHOD IN COMBINATORICS. Theorem 1.1 (Oddtown theorem). In a town of n citizens, no more than n clubs can be formed under the rules

Size: px

Start display at page:

Download "LINEAR ALGEBRA METHOD IN COMBINATORICS. Theorem 1.1 (Oddtown theorem). In a town of n citizens, no more than n clubs can be formed under the rules"

Edward Daniel
5 years ago
Views:

1 LINEAR ALGEBRA METHOD IN COMBINATORICS 1 Warming-up example Theorem 11 (Oddtown theorem) In a town of n citizen, no more tha club can be formed under the rule each club have an odd number of member each pair of club hare an eveumber of member Proof It i enough to how that the incidence vector v i are linearly independent over F 2 To prove thi, one jut oberve that v i, v j = δ ij over F 2 Theorem 12 The ame concluion hold if we revere the rule: each club have an eveumber of member each pair of club hare an odd number of member Let a 1,, a m be point in R n, it i clear that if all the pairwie ditance d(a i, a j ) are equal, then m n + 1 Now aume that d(a i, a j ) can take two value, then how big m can be? Theorem 13 (Two ditance) Let m(n) be the larget number m can take, then one ha n(n + 1)/2 m(n) (n + 1)(n + 4)/2 Proof The lower bound can be obtained by conidering (,,,, 1,,,, 1,,, ) For the upper bound, aume that the ditance take value δ 1 and δ 2 For each 1 i m, we define the polynomial f i : R n R: f i (x) := ( x a i 2 2 δ 2 1)( x a i 2 2 δ 2 2) Notice that f i (a i ) and f i (a j ) = for all j i Becaue of thi, f i (x) are linearly independent over the linear pace generated by {( n k=1 x2 k )2, ( n k=1 x2 k )x j, x i x j, x i, 1} Similarly, one alo ha 1

2 2 LINEAR ALGEBRA METHOD IN COMBINATORICS Theorem 14 (-ditance) (n ) + 1 m(n, Our next example i on the decompoition of K n into complete bipartite graph Theorem 15 If the edge et of the complete graph o vertice i the dijoint union of the edge et of m complete bipartite graph, then m n 1 ) Proof For each complete bipartite graph (X k, Y k ), we aign a n matrix A k in which a ij = 1 if and only if i X k and j Y k It i clear that S = A k ha the property that S + S T = J I We next claim that r(s) n 1 Indeed, otherwie there exit x = (x 1,, x n ) with x x n = and Sx = Thu S T x = x, and o = x T S T x = x 2 Here i another example uing more pectral propertie Theorem 16 Aume that G i r regular graph with r vertice and with girth 5, then r {1, 3, 5, 7, 57} Proof One firt oberve that for any vertex-pair (v 1, v 2 ), N(v 1 ) N(v 2 ) = 1 Thi motivate u to conider the adjacency matrix A It ha the following propertie: A 2 = ri + Ā I + A + Ā = J From here, it i not hard to how that either r = 2 or (m 1 m 2 ) = 15 with r = ( 2 + 3)/4 Thu r {1, 3, 7, 57} 2 Set ytem with retricted interection Theorem 21 (Non-uniform Fiher inequality, Majumdar 1953) Let C 1,, C m be ditinct ubet of [n] uch that for every i j, C i C j = λ for ome 1 λ n Then m n Proof If uffice to aume that C i > λ for all i Let M of ize m n be the incidence matrix of our ytem, then MM T = λj + C, where C i a diagonal matrix of poitive entrie It i eay to check that MM T i poitive definite, and thu M mut have full rank Remark 22 Fiher original reult (194) wa for λ = 1 together with uniformity auming on the ize of C i Thi uniformity condition wa then relaxed by Erdo and de Bruijn (1948), and generalized by Boe a year later by a linear algebraic method argument

3 LINEAR ALGEBRA METHOD IN COMBINATORICS 3 It i perhap ueful to ummarize the algebraic technique we have ued o far: Propoition 23 (Matrice If M i poitive-definite, then it ha full rank If A, B ha ize a b and b a repectively, and if rank(ab) = a, then a b If A, B are matrice of the ame ize, then rank(a + B) rank(a) + rank(b) Spectral decompoition for normal matrice (ymmetric matrice Propoition 24 (Criteria for linear independence) Let f i : Ω F, 1 i m be function Then they are linearly independent over F if one of the following hold for ome a i Ω, 1 i m (diagonal) f i (a j ) = if i j and if i = j, (triangular) f i (a j ) = if i < j and if i = j Theorem 25 (Frankl-Wilon 1981) Let L be a et of integer and F an L-interecting family of ubet of [n] Then Proof Arrange the et ize in increaing order A 1 A 2 A m Let v i be the indicator function of A i, define f i : R n R a follow: f i (x) = l k < A i (x v i l k ) Notice that f i ha degree at mot, and f i (v i ) while f i (v j ) = for j i Deform f i to multilinear f i uch that f i = f i over all v i A f i are linearly independent by the triangular criterion, m i By the ame method, one can obtain the following modulo verion Theorem 26 Let p be a prime number, and L be a et of integer F = {A 1,, A m } uch that Aume that A i / L( mod p); A i A j L( mod p) Then A a corollary, one deduce that

4 4 LINEAR ALGEBRA METHOD IN COMBINATORICS Corollary 27 Let p be a prime and F a (2p 1)-uniform family of ubet of a et of 4p 1 element If no two of F interect in preciely p 1 element, then ( ) 4p 1 2 p 1 Proof Let L := {,, p 2} and ue + + ) 2 Corollary 28 Let L be a et of integer and F an L-interecting k-uniform family of ubet of a et of n element, where k Then The following celebrated reult of Ray-Chaudhuri and Wilon how that the bound can be improved to Theorem 29 (Ray-Chaudhuri -Wilon 1975) Let L be a et of integer and F an L- interecting k-uniform family of ubet of a et of n element, where k Then Proof (of Theorem 29) One how that the f i (from the proof of Theorem 25) together with x I (x 1 + +x n k), I 1 are independent Indeed, a vanihing linear combination m i=1 λ if i + I 1 µ Ix I (x x n k) would imply λ i = (by replacing v i into the indentity) For I 1 µ Ix I (x x n k), one ha x I (x x n k)(v J ) = if J I, J I, and if J = I We now preent an application of Corollary 27 Theorem 21 [Chromatic number of unit ditance graph] For large n, the chromatic number of the unit ditance graph on R n i greater than 113 n Notice that the upper bound i at mot /2 Proof (of Theorem 21) Without lo of generality, aume that n = 4p 1 and define a graph G on ( [n] 2p 1) by connecting A to B if A B = p 1 Notice that d(a, B) = 2p in thi cae A any independent et of G ha ize at mot 2 ( ) 4p 1 p 1 according to Corollary 27 The chromatic number of G i at leat ( ) ( 4p 1 2p 1 / 4p 1 ) p p 1 Theorem 211 (Kahn-Kalai diproving of Boruk conjecture) Let f(n) denote the minimum number uch that every et of diameter 1 in R d can be partitioned into f(d) piece of maller diameter Then f(d) 12 d

5 LINEAR ALGEBRA METHOD IN COMBINATORICS 5 Proof For each A V (G) from the proof of Theorem 21, we define Φ(A) 2 X with X = ( ) [n] 2 a follow: One check that φ(a) = {{x, y} : x A, y Ā} (1) F = {φ(a) : A V (G)} i k(n k) uniform, with k = 2p 1 (2) Aume that A B = r, then φ(a) φ(b) = r(n 2k + r) + (k r) 2, which i minimized when r i a cloe to k n/4 = p 3/4 a poible, ie when r = p 1 It follow from (2) that G and φ(g) i iomorphic, where φ(g) i the graph over φ(a), A V (G), and φ(a) i connected to φ(b) if their ditance i the diameter µ(f) of F By the proof of Theorem 21, with d = 2), χ(φ(g)) 113 n = 113 2d Reference [1] L Babai and P Frankl, Linear Algebra method in Combinatoric

List Coloring Graphs

List Coloring Graphs Lit Coloring Graph February 6, 004 LIST COLORINGS AND CHOICE NUMBER Thomaen Long Grotzch girth 5 verion Thomaen Long Let G be a connected planar graph of girth at leat 5. Let A be a et of vertice in G